diff --git "a/OtAzT4oBgHgl3EQfIftM/content/tmp_files/load_file.txt" "b/OtAzT4oBgHgl3EQfIftM/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/OtAzT4oBgHgl3EQfIftM/content/tmp_files/load_file.txt" @@ -0,0 +1,1166 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf,len=1165 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='01062v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='AT] 3 Jan 2023 ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II OSCAR RANDAL-WILLIAMS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We describe the ring structure of the rational cohomology of the Torelli groups of the manifolds #gSn × Sn in a stable range, for 2n ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Some of our results are also valid for 2n = 2, where they are closely related to unpublished results of Kawazumi and Morita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Characteristic classes 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Twisted cohomology of diffeomorphism groups 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Cohomology of Torelli groups 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The case 2n = 2 22 References 28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Introduction This paper can be considered as a (somewhat extensive) addendum to our earlier work with Kupers [KRW20b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We shall be concerned with the manifold Wg := #gSn × Sn generalising to higher dimensions the orientable surface of genus g, its topological group Diff+(Wg) of orientation-preserving diffeomorphisms, and various subgroups of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The first kind of subgroups are Diff(Wg, D2n) ≤ Diff+(Wg, ∗) ≤ Diff+(Wg), the diffeomorphisms which fix a disc and a point respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The second kind are their Torelli subgroups Tor(Wg, D2n), Tor+(Wg, ∗), Tor+(Wg), consisting of those diffeomorphisms which in addition act trivially on Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The intersection form on this middle cohomology group is nondegenerate and (−1)n- symmetric, giving a homomorphism αg : Diff+(Wg) −→ Gg := � Sp2g(Z) if n is odd, Og,g(Z) if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This map is not always surjective, but its image is a certain finite-index subgroup G′ g ≤ Gg, even when restricted to Diff(Wg, D2n), so there is an outer G′ g-action on each of the Torelli subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This makes the rational cohomology of each of the Torelli groups into G′ g-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In [KRW20b], for 2n ≥ 6 we determined H∗(BTor(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) as a ring and as a G′ g-representation in a range of degrees tending to infinity with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using the Serre spectral sequence associated to various simple fibrations relating the dif- ferent Torelli groups we were able to also determine H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) and H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) as G′ g-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This kind of argument was not able to 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 55R40, 11F75, 57S05, 18D10, 20G05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Cohomology of diffeomorphism groups, Torelli groups, cohomology of arithmetic groups, Miller-Morita-Mumford classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1 2 OSCAR RANDAL-WILLIAMS determine the ring structure, however, as multiplicative information gets lost when passing to the associated graded of the Serre filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Here we shall determine H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) and H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) as Q-algebras too: this is achieved in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The statement given there is more powerful, but just as in [KRW20b, Section 5] one can extract from it the following presentation for H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q), which is easier to parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (A presentation for H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) can be extracted in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=') Let us write H(g) := Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q), on which G′ g operates in the evident way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let λ : H(g) ⊗ H(g) → Q denote the intersection form, and {ai}2g i=1 be a basis of H(g) with dual basis {a# i }2g i=1 in the sense that λ(a# i , aj) = δij, so that the form dual to the pairing λ is ω = �2g i=1 ai ⊗ a# i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2 we will construct certain “modified twisted Miller–Morita–Mumford classes”, which when restricted to the Torelli group yield G′ g-equivariant maps ¯κc : H(g)⊗r −→ Hn(r−2)+|c|(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) for each c ∈ Q[e, p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , pn−1] = H∗(BSO(2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) and each s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' When r = 0 we write ¯κc = ¯κc(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' these agree with the usual Miller–Morita–Mumford classes κc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If 2n ≥ 6 then, in a range of degrees tending to infinity with g, H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is generated as a Q-algebra by the classes ¯κc(v1 ⊗ · · · ⊗ vr) for c a monomial in e, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , pn−1, and r ≥ 0, such that n(r −2)+|c| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' A complete set of relations in this range is given by (i) ¯κc(vσ(1) ⊗ · · · ⊗ vσ(r)) = sign(σ)n · ¯κc(v1 ⊗ · · · ⊗ vr), (ii) ¯κe(v1) = 0, (iii) � i ¯κx(v ⊗ ai) · ¯κy(a# i ⊗ w) = ¯κx·y(v ⊗ w) + 1 χ2 ¯κe2 · ¯κx(v) · ¯κy(w) − 1 χ � ¯κe·x(v) · ¯κy(w) + ¯κx(v) · ¯κe·y(w) � , (iv) � i ¯κx(v ⊗ ai ⊗ a# i ) = χ−2 χ ¯κe·x(v) + 1 χ2 ¯κe2 · ¯κx(v), (v) ¯κLi = 0, for v ∈ H(g)⊗r and w ∈ H(g)⊗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For 2n = 4 or 2n = 2 there is still a map from the Q-algebra given by this presentation to H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If 2n = 2 then (in a stable range) this map is an isomorphism onto the maximal algebraic subrepresentation in degrees ≤ N, assuming that H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is finite-dimensional in degrees < N for all large enough g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This is known to hold for N = 2 by work of Johnson [Joh85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The overall strategy is parallel to [KRW20b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There we defined cer- tain twisted Miller–Morita–Mumford classes and used them to describe the twisted cohomology groups H∗(BDiff+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗s) in a stable range of degrees, where H is the local coefficient system corresponding to Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) with the action by dif- feomorphisms of Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This calculation was valid for 2n = 2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using that for 2n ≥ 6 the G′ g-representations H∗(BTor+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) extend to representations of the ambient algebraic group (namely Sp2g or Og,g) by [KRW20a]1, the argument was completed by establishing the degeneration of the Serre spectral sequence Ep,q 2 = Hp(G′ g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Hq(BTor+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)⊗H⊗s) =⇒ Hp+q(BDiff+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗s) using work of Borel, and then using a categorical form of Schur–Weyl duality to ex- tract the structure of H∗(BTor+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) from the H∗(BDiff+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗s) for all s’s and various structure maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1In fact we did something more complicated in [KRW20b] because this algebraicity result was not known at the time, but please allow some narrative leeway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 3 The twisted Miller–Morita–Mumford classes may be defined on BDiff+(Wg, ∗) too, but not on BDiff+(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Our first task will be to define so-called “modi- fied twisted Miller–Morita–Mumford classes” in H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗s) and analyse their behaviour: it turns out that their behaviour is significantly more complicated than the unmodified version, though still understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We will then use them to describe the twisted cohomology groups H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗s) in a stable range of degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This description will be in terms of a certain vector space of graphs with vertices labelled by monomials in Euler and Pontrjagin classes, which play the role here of the vector spaces of labelled partitions from [KRW20b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The passage from this calculation to H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The case of dimension 2n = 2 is somewhat special, in precisely the same way as it was in [KRW20b]: the calculation of H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗s) is valid in this case, but as the cohomology of BTor+(Wg) is not even known to be finite-dimensional in a stable range, we cannot make a conclusion about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (Instead one can make a conclusion about the continuous cohomology of the Torelli group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' the Lie algebra cohomology of its Mal’cev Lie algebra: see [KRW21], [FNW21], [Hai20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=') In addition, in this case our modified twisted Miller–Morita–Mumford classes are essentially the same as those that have been defined by Kawazumi and Morita [Mor96, KM96, KM01], and the graphical calculus that we employ is similar to theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In Section 5 we fully explain this connection, and also relate it to work of Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To avoid a great deal of repetition we have refrained from spelling out a lot of the background that was given in [KRW20b], and from giving in detail arguments that are very similar to those given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As such this paper should not be considered as attempting to be self-contained: given that its interest will be to readers of [KRW20b] this should not present a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' I am grateful to Alexander Kupers for feedback on an earlier draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' I was supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 756444) and by a Philip Leverhulme Prize from the Leverhulme Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Characteristic classes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Recollection on twisted Miller–Morita–Mumford classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If π′ : E′ → X′ is an oriented smooth W 2n g -bundle equipped with a section s : X′ → E′, and H denotes the local coefficient system x �→ Hn((π′)−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) on X′, then it is explained in [KRW20b, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2] that there is a unique class ε = εs ∈ Hn(E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) characterised by (i) for each x ∈ X′ the element ε|(π′)−1(x) ∈ Hn((π′)−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)⊗Hn((π′)−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is coevaluation, and (ii) s∗ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The proof is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The Serre spectral sequence yields an exact sequence 0 → Hn(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) (π′)∗ → Hn(E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) → H0(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H∨⊗H) dn+1 → Hn+1(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) (π′)∗ → Hn+1(E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) and the section s shows that the right-hand map (π′)∗ is injective, so that the map dn+1 is zero, and splits the left-hand map (π′)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The class coev ∈ H0(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H∨ ⊗ H) then gives rise to a unique ε satisfying the given properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We then defined the twisted Miller–Morita–Mumford class (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) κεac = κεac(π′, s) := � π′ εa · c(Tπ′E′) ∈ H(a−2)n+|c|(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 4 OSCAR RANDAL-WILLIAMS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Modified twisted Miller–Morita–Mumford classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If π : E → X is an oriented smooth W 2n g -bundle but is not equipped with a section then, as long as χ := χ(Wg) = 2 + (−1)n2g ̸= 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (n, g) ̸= (odd, 1), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1), the cohomological role of the section can instead be played by the transfer map 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e · −) : H∗(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) −→ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H), where e := e(TπE) ∈ H2n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) denotes the Euler class of the vertical tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The projection formula 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e · π∗(x)) = 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e) · x = χ χx = x shows that this map splits π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus in this situation there is a unique class ¯ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) characterised by (i) for each x ∈ X the element ¯ε|π−1(x) ∈ Hn(π−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ Hn(π−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is coevaluation, and (ii) 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e · ¯ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If (n, g) = (odd, 1) then there is no class ¯ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) satisfying (i) and natural under pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To see this it suffices to give one example of a smooth oriented W1-bundle for which ¯ε does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider the Borel construction for the evident action of S1 × S1 on W1 = Sn × Sn given by considering Sn as the unit sphere in C(n+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This gives a smoth oriented W1-bundle over B(S1 × S1) with total space E ≃ CP(n−1)/2 × CP(n−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) = 0 as n is odd but E has a cell structure with only even-dimensional cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By analogy with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) we may then define the modified twisted Miller–Morita– Mumford class (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2) κ¯εac = κ¯εac(π) := � π ¯εa · c(TπE) ∈ H(a−2)n+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If π : E → X does have a section s : X → E then the class ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) is also defined, and we may compare it with ¯ε as follows: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If π : E → X has a section then ¯ε = ε − 1 χπ∗κεe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The classes ε, ¯ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) are both defined, and agree when restricted to the fibres of the map π, so by considering the Serre spectral sequence for π we must have ¯ε − ε = π∗(x) for some class x ∈ Hn(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Applying 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e · −) we see that x = 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e · (¯ε − ε)) = 0 − 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e · ε) = − 1 χκεe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (Here we have used, as we often will, the fact that e has even degree to commute it past ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=') □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 (Splitting principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The pullback (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3) E1 ×X E2 E2 E1 X, pr1 pr2 π2 π1 where πi : Ei → X are copies of the map π, is equipped with a section given by the diagonal map ∆ : E1 → E1 ×X E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As the maps π∗ 1 and pr∗ 2 are injective (they are split by their corresponding transfer maps), for the purpose of establishing identities between the characteristic classes we have discussed it suffices to do so for bundles which do have a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There is another description of ¯ε which is sometimes useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let pr1 : E ×X E → E be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3), which is an oriented Wg-bundle with section given by the diagonal map ∆, and so has the class κεe(pr1, ∆) defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We have ¯ε = − 1 χκεe(pr1, ∆) ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 we may suppose without loss of generality that π : E → X has a section s : X → E, defining a class ε = εs ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider the pullback square (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' let ei = (pri)∗(e) ∈ H2n(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) be the Euler class of the vertical tangent bundle on the ith factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Considering pr1 as a Wg-bundle with section given by the diagonal map ∆, there is a class ε∆ ∈ Hn(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As both ε∆ and pr∗ 2(εs) restrict to coevaluation on the fibres of pr1, we have ε∆ − pr∗ 2(εs) = pr∗ 1(x) for some class x ∈ Hn(E1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Pulling this equation back along ∆ shows that x = −εs, so ε∆ = pr∗ 2(εs) − pr∗ 1(εs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Then we have κεe(pr1, ∆) = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (ε∆ · e2) = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ((pr∗ 2(εs) − pr∗ 1(εs)) · e2) = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (pr∗ 2(εs · e)) − (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (pr∗ 1(εs) · e2) = π∗ 1(π2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (εs · e) − χεs = π∗ 1κεe(π, s) − χεs = −χ¯ε as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ The intersection form of the fibres of π : E → X provides a map of local coeffi- cient systems λ : H⊗H → Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' as we will often be concerned with applying it to two factors of a tensor power H⊗k and will have to specify which factors we apply it to, we will denote λ by λ1,2 and more generally write λi,j : H⊗k → H⊗k−2 for the map that applies λ to the ith and jth factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We call such operations contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If p : E1 ×X E2 → X denotes the fibre product of two copies of π : E → X, and if this has a section s : X → E, then in [KRW20b, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='9] we have established the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4) λ1,2(ε × ε) = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (1) − 1 × v − v × 1 + p∗s∗e ∈ H2n(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q), where v = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (1) ∈ H2n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is the fibrewise Poincar´e dual to the section s, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [KRW20b, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The analogue of this formula for ¯ε is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We have λ1,2(¯ε × ¯ε) = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (1) + 1 χ2 p∗κe2 − 1 χ(e × 1 + 1 × e) ∈ H2n(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 we may suppose without loss of generality that π : E → X has a section s : X → E, so that ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2 we have ¯ε = ε − 1 χπ∗κεe ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H), and so λ1,2(¯ε × ¯ε) = λ1,2((ε − 1 χπ∗κe·ε) × (ε − 1 χπ∗κe·ε)) = λ1,2(ε × ε) − λ1,2( 1 χπ∗κεe × ε) − λ1,2(ε × 1 χπ∗κεe) + λ1,2( 1 χπ∗κεe × 1 χπ∗κεe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The first term is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4), and using [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='10] the last term is given by λ1,2( 1 χπ∗κεe × 1 χπ∗κεe) = 1 χ2 p∗λ1,2(κεe · κεe) = 1 χ2 p∗(κe2 + (χ2 − 2χ)s∗e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For the middle two terms, note that ε × 1 χπ∗κεe = 1 χ(ε × 1) · p∗(κe·ε) = 1 χ(ε · π∗κεe) × 1 6 OSCAR RANDAL-WILLIAMS so we need to calculate λ1,2(ε · π∗κεe) ∈ H2n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The class ε · κεe is the fibre integral along pr1 : E1 ×X E2 → E1 of ε × (ε · e) = (ε × ε) · (1 × e), so λ1,2(ε · κεe) = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (λ1,2(ε × ε) · (1 × e)) = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='((∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (1) − 1 × v − v × 1 + p∗s∗e) · (1 × e)) = e − π∗s∗e − χv + χπ∗s∗e and hence λ1,2(ε × 1 χκεe) = 1 χ(e − π∗s∗e − χv + χπ∗s∗e) × 1 = 1 χe × 1 + χ−1 χ p∗s∗e − v × 1 and similarly λ1,2( 1 χκεe × ε) = 1 χ1 × e + χ−1 χ p∗s∗e − 1 × v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Combining these gives λ1,2(¯ε × ¯ε) = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (1) − 1 × v − v × 1 + p∗s∗e + 1 χ2 p∗κe2 + χ−2 χ p∗s∗e − ( 1 χe × 1 + χ−1 χ p∗s∗e − v × 1) − ( 1 χ1 × e + χ−1 χ p∗s∗e − 1 × v) = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (1) + 1 χ2 p∗κe2 − 1 χ(e × 1 + 1 × e) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ If in addition we have a lift ℓ : E → B of the fibrewise Gauss map along some fibration θ : B → BSO(2n) then for any c ∈ H∗(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) we can define modified twisted Miller–Morita–Mumford classes by the formula κ¯εac := π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Under the action of a permutation σ ∈ Sa of the tensor factors these classes transform as sign(σ)n, as ¯ε has degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus for any finite set S there is a well-defined element (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5) κ¯εSc := π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) ⊗ (det QS)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To keep track of signs, for an ordered set S = {s1 < s2 < · · · < sa} we will often write κ¯εs1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',sac ∈ Hn(a−2)+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) for the corresponding element, understand- ing that if σ is a reordering of S then κ¯εσ(s1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',σ(sa)c = sign(σ)nκ¯εs1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',sa c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5 we immediately see that these characteristic classes satisfy the following analogue of the contraction formula from [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='6 (Modified contraction formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−) we have the identities λ1,2(π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a · ℓ∗c)) = ( χ−2 χ )π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε3,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a · ℓ∗(e · c)) + 1 χ2 κe2 · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε3,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a · ℓ∗c) and λa,a+1(π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a · ℓ∗c) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯εa+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a+b · ℓ∗c′)) = π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a−1,a+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a+b · ℓ∗(c · c′)) + 1 χ2 κe2 · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a−1 · ℓ∗c) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯εa+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a+b · ℓ∗c′) − 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a−1 · ℓ∗(e · c)) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯εa+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a+b · ℓ∗c′) − 1 χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a−1 · ℓ∗c) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (¯εa+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',a+b · ℓ∗(e · c′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Similarly, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2 we immediately obtain the following: ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If the bundle π : E → X has a section, so that the class ε and hence κεSc is defined, then κ¯εSc = � I⊆S κεIc(− 1 χκεe)S\\I ∈ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) ⊗ (det QS)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let us give an example of using the modified contraction formula to evaluate an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider the class λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Then λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = λ1,5λ2,6 � κ¯ε1,2,5,6 + 1 χ2 κe2κ¯ε1,2κ¯ε5,6 − 1 χ(κ¯ε1,2eκ¯ε5,6 + κ¯ε1,2κ¯ε5,6e) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The first term is λ1,5λ2,6(κ¯ε1,2,5,6) = (−1)nλ1,5λ2,6(κ¯ε1,5,2,6) = (−1)nλ2,6( χ−2 χ κ¯ε2,6e + 1 χ2 κe2κ¯ε2,6) = (−1)n χ−2 χ ( χ−2 χ κe2 + 1 χ2 κe2χ) + (−1)n 1 χ2 κe2( χ−2 χ χ) = (−1)n( (χ−2)2 χ2 + 2 χ−2 χ2 )κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The second term is 1 χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε5,6) = (−1)n 1 χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε6,5) = (−1)n 1 χ2 κe2λ1,5(κ¯ε1,5) = (−1)n χ−2 χ2 κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The third term is − 1 χλ1,5λ2,6(κ¯ε1,2eκ¯ε5,6) = (−1)n+1 1 χλ1,5λ2,6(κ¯ε1,2eκ¯ε6,5) = (−1)n+1 1 χλ1,5(κ¯ε1,5e − 1 χκ¯ε1eκ¯ε5e) = (−1)n+1 1 χ � ( χ−2 χ κe2 + 1 χ2 κe2χ) − 1 χ(κe2 + 1 χ2 κe2χ2 − 1 χ(2χκe2)) � = (−1)n+1( χ−2 χ2 + 1 χ2 − 1 χ2 − 1 χ2 + 2 χ2 )κe2 = (−1)n+1 χ−1 χ2 κe2 and the fourth term is the same as the third by the evident symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In total we have λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = (−1)n( (χ−2)2 χ2 + 2 χ−2 χ2 + χ−2 χ2 − 2 χ−1 χ2 )κe2 = (−1)n χ−3 χ κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Graphical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In [KRW20b, Section 5] it was found to be very convenient to adopt a graphical formalism where κεac corresponds to a vertex with a half-edges incident to it and a formal label c, a product of κεac’s corresponds to a disjoint union of such vertices, and applying the contraction λi,j corresponds to pairing up the half-edges labelled i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It will be convenient to adopt a similar formalism here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let S be a finite set, and V be a graded Q-algebra with a distinguished element e ∈ V2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Slightly modifying2 2The difference is that we allow labelled vertices whose contribution to the degree is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 8 OSCAR RANDAL-WILLIAMS the definition from [KRW20b, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1], a marked oriented graph with legs S and labelled by V consists of the following data: (i) a totally ordered finite set ⃗V (of vertices), a totally ordered finite set ⃗H (of half-edges), and a monotone function a: ⃗H → ⃗V (encoding that a half-edge h is incident to the vertex a(h)), (ii) an ordered matching m = {(ai, bi)}i∈I of the set H ⊔ S (encoding the oriented edges of the graph), (iii) a function c: V → V with homogeneous values, such that |c(v)|+n(|a−1(v)|− 2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Marked oriented graphs Γ = (⃗V , ⃗H, a, m, c) and Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) with the same set of legs S are isomorphic if there are order-preserving bijections ⃗V ∼ → ⃗V ′ and ⃗H ∼ → ⃗H′ which intertwine a and a′, intertwine c and c′, and send m to m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' An oriented graph is an isomorphism class [Γ] of marked oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We assign to a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) the degree deg(Γ) := � v∈V � |c(v)| + n(|a−1(v)| − 2) � = n(|H| − 2|V |) + � v∈V |c(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let π : E → X be an oriented Wg-bundle with a lift ℓ : E → B of the map clas- sifying the vertical tangent bundle along θ : B → BSO(2n), and let V := H∗(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) and e := θ∗e ∈ V2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Then given a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) with legs S we form a class ¯κ(Γ) ∈ Hdeg(Γ)(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) by the following recipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Firstly, we may form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='6) � v∈V κ¯εa−1(v)c(v) ∈ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗H), where we have used the ordering on ⃗V to order the product, and the ordering on ⃗H to trivialise the factor of (det QH)⊗n = (� v∈V det Qa−1(v))⊗n that arises from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Secondly, taking two copies S1 and S2 of the set S and writing si ∈ Si for the element corresponding to s ∈ S we can form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7) � s∈S κ¯εs1,s2 ∈ H∗(X, H⊗(S1⊔S2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As each κ¯εs1,s2 has degree 0, the product does not depend on how the factors are ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Taking the product of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7) and then applying λx,y for each ordered pair (x, y) in the matching m on H ⊔ S = H ⊔ S1 gives the required class ¯κ(Γ) ∈ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S2) = H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In this graphical interpretation we recognise the class evaluated in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='8 as that associated to the theta-graph with a certain ordering and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Clearly ¯κ(Γ) only depends on the underlying oriented graph [Γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We now describe how it transforms when the orderings on V , H, and the pairs m are changed, without changing the underlying labelled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) is another marked oriented graph and there are bijections f : H → H′ and g : V → V ′ intertwining a and a′ and c and c′ and such that under these bijections the matching m′ differs from m by reversing the order of k pairs, then ¯κ(Γ′) = (−1)nksign(f)sign(g) · ¯κ(Γ) for certain signs described on [KRW20b, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 55-56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Graphs considered as representing ¯κ’s behave differently to those representing κ’s described in [KRW20b, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To distinguish them we will depict the graphs ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 9 representing κ’s in red, as we did in that paper, and the graphs representing ¯κ’s in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The contraction formula of [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='10] was interpreted in [KRW20b, Section 5] as giving relations among red graphs which yield equivalent κ-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In the generality of a smooth oriented Wg-bundle π : E → X with section s : X → E these may be depicted as follows: c = ce +s∗e c −2s∗c c c′ = cc′ +s∗e c c′ −s∗c c′ −s∗c′ c Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The contraction formula, displayed graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Here the negative terms only arise when they make sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' when the vertex has valence 2 in the first case, when the vertex labelled c has valence 1 in the second case, and when the vertex labelled c′ has valence 1 in the third case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In these and the following figures, to avoid clutter we have adopted the following ordering conventions: vertices are numbered starting from 1 from left to right, half-edges around each vertex are ordered clockwise starting from the marked half-edge, and edges are oriented from the smaller half-edge to the larger one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Similarly, the modified contraction formula of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='6 can be interpreted as giving the following relations among blue graphs which yield equivalent ¯κ-classes: c = χ−2 χ ce + 1 χ2 c e2 c c′ = cc′ + 1 χ2 c c′ e2 − 1 χ( ce c′ + c c′e ) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The modified contraction formula, displayed graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Twisted cohomology of diffeomorphism groups The main goal of this section is to describe the twisted cohomology groups H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) and H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) in a stable range of degrees, of the classifying space BDiff+(Wg) of the group of orientation-preserving diffeomorphisms of Wg (which classifies oriented Wg-bundles), and the classifying space BDiff+(Wg, ∗) of the group of orientation-preserving dif- feomorphisms of Wg which fix a point ∗ ∈ Wg (which classifies oriented Wg-bundles with section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15] the analogous calculation was given for 10 OSCAR RANDAL-WILLIAMS the classifying space BDiff(Wg, D2n) of the group of diffeomorphisms of Wg which fix a disc D2n ⊂ Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In order to do this we will also discuss the manifolds Wg equipped with θ- structures for the tangential structure θ : BSO(2n)⟨n⟩ → BO(2n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' the n- connected cover of BO(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In this case we will consider the homotopy quotients BDiffθ(Wg) := Bun(T Wg, θ∗γ2n)//Diff(Wg) BDiffθ(Wg, ∗) := Bun(T Wg, θ∗γ2n)//Diff(Wg, ∗) where Bun(T Wg, θ∗γ2n) denotes the space of vector bundle maps T Wg → θ∗γ2n from the tangent bundle of Wg to the bundle classified by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The group Diff(Wg) acts on the space of bundle maps by precomposing with the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There is a factorisation θ : BSO(2n)⟨n⟩ θor → BSO(2n) σ→ BO(2n), and by ob- struction theory one sees that the space Bun(T Wg, σ∗γ2n) has two contractible path components corresponding to the two orientations of Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In particular there are equivalences Bun(T Wg, σ∗γ2n)//Diff(Wg) ≃ BDiff+(Wg) Bun(T Wg, σ∗γ2n)//Diff(Wg, ∗) ≃ BDiff+(Wg, ∗) and so θor induces maps BDiffθ(Wg) −→ BDiff+(Wg) and BDiffθ(Wg, ∗) −→ BDiff+(Wg, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It is shown in [GRW19, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2] that these are principal SO[0, n − 1]-fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In particular the spaces BDiffθ(Wg) and BDiffθ(Wg, ∗) are path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Spaces of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Our description of the twisted cohomology groups of BDiff+(Wg), BDiff+(Wg, ∗), BDiff(Wg, D2n), BDiffθ(Wg) and BDiffθ(Wg, ∗) in a stable range will be—via the graphical interpretation given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3—in terms of graded vector spaces of labelled graphs, modulo certain relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (Readers of [KRW20b] may have been expecting vector spaces of labelled partitions instead: here we have found spaces of graphs more convenient for formulating results, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2, though spaces of labelled partitions will still play a role in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=') To describe these spaces of graphs we will use the graded Q-algebras V := H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) = Q[p⌈ n+1 4 ⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , pn−1, e] W := H∗(BSO(2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) = Q[p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , pn−1, e] with distinguished elements e of degree 2n given by the Euler class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In order to work in a way which is agnostic about the genus g of the manifold Wg under consideration, we will work over the ring Q[χ±1] instead of Q, where χ is an invertible formal parameter which will—later—be set to the Euler characteristic 2 + (−1)n2g of Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (i) Let Graph1(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼ where ∼ (a) imposes the sign rule for changing orderings of vertices and half-edges and for reversing orientations of edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (b) imposes linearity in the labels, and sets a graph containing an a-valent vertex labelled by c with |c| + n(a − 1) < 0 to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (c) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, and if 2n ≡ 0 mod 4 sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 11 (d) imposes the contraction relations3 c = ce − 2c c c′ = cc′ − c c′ − c′ c where the negative terms only arises when they makes sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' in the first case when the vertex has valence 2 and its label c is a scalar multiple of 1 ∈ V0, in the second case when c is a scalar multiple of 1 ∈ V0 and has valence 1, and similarly in the third case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (ii) Let Graphθ ∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V} ⊗ V/ ∼ where ∼ imposes (a)–(c) as well as (d′) imposes the contraction relations of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (iii) Let Graph∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W} ⊗ W/ ∼ where ∼ imposes (a) and (b), as well as (c′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent vertex labelled by any degree 2n monomial in Pontrjagin classes equal to 0, and for any 1 ≤ i ≤ ⌊n/4⌋ sets cpi = 1 χ c ⊗pi and (d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (iv) Let Graphθ(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼ where ∼ imposes (a) and (b), as well as (c′′′) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, if 2n ≡ 0 mod 4 sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0, and sets the 1-valent vertex labelled by e ∈ V2n equal to 0, (d′′′) imposes the contraction relations of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (v) Let Graph(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W}/ ∼ where ∼ imposes (a), (b), as well as (c′′′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent vertex labelled by any degree 2n monomial in Pontrjagin classes equal to 0, sets the 1-valent vertex labelled by e ∈ W2n equal to 0, and for any 1 ≤ i ≤ ⌊n/4⌋ sets 3These are the relations from Figure 1 when s∗ kills all positive-degree classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 12 OSCAR RANDAL-WILLIAMS cpi = 1 χ c epi and (d′′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2 (Graphs and partitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In all cases one can apply the (modified) contraction formula to pass from a graph to a sum of graphs with strictly fewer edges, and so by iterating to a sum of graphs with no edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' These are disjoint unions of labelled corollas, and so correspond to partitions of S with labels in V or W, plus additional external labels in cases (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There are two issues with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The first is that in cases (iv) and (v) it is not clear that the resulting sum of disjoint unions of labelled corollas is unique, as one has to choose an order in which to eliminate edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The second is that even if it is, then the functoriality on the Brauer category which we describe below would involve gluing in edges and then eliminating them, leading to a complicated formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This is why we have found it convenient to work with spaces of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We wish to consider each of the above as defining functors on the (signed) Brauer category as in [KRW20b, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3], but to take into account the parameter χ we must slightly generalise to a Q[χ]-linear version of the (signed) Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For finite sets S and T let preBrχ(S, T ) be the free Q[χ]-module on tuples (f, mS, mT ) of a bijection f from a subset S◦ ⊂ S to a subset T ◦ ⊂ T , an ordered matching mS of S \\ S◦, and an ordered matching mT of T \\ T ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let Brχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) − (f, m′ S, m′ T ) whenever mS agrees with m′ S after reversing some pairs, and mT agrees with m′ T after reversing some pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let sBrχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) − (−1)kl(f, m′ S, m′ T ) whenever mS agrees with m′ S after reversing k pairs, and mT agrees with m′ T after reversing l pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let (s)Brχ be the Q[χ]-linear category whose objects are finite sets, and whose morphisms are the Q[χ]-modules (s)Brχ(S, T ) defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In the case of Brχ we think of [f, mS, mT ] as representing 1-dimensional cobordisms with no closed components: then the composition law is given by composing cobordisms and then replacing each closed 1-manifold by a factor of χ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In the case of sBrχ we think of (f, mS, mT ) as representing oriented 1-dimensional cobordisms with no closed components: then the composition law is given by composing cobordisms and then replacing each compatibly oriented closed 1-manifold by a factor of −(χ − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let d(s)Brχ denote the subcategories having all objects and morphisms spanned by [f, mS, mT ] with T ◦ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For a central charge d ∈ Q let (d)(s)Brd denote the Q-linear category obtained by specialising the Q[χ]-linear category (d)(s)Brχ to χ = 2+d for (d)Br or χ = 2−d for (d)sBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (This notation then agrees with [KRW20b, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=') We consider the spaces of graphs above as defining Q[χ]-linear functors Graph1(−), Graphθ ∗(−), Graph∗(−), Graphθ(−), Graph(−) : (s)Brχ → Gr(Q[χ±1]-mod) in the evident way, by gluing of oriented graphs (after orientations have been ar- ranged to be compatible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We endow them with a lax symmetric monoidality by disjoint union of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We write Graph1(−)g : (s)Br2g → Gr(Q-mod) and so on for their specialisations at χ = 2 + (−1)n2g (defined for (n, g) ̸= (odd, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The isomorphism theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 below extends [KRW20b, Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15] to BDiffθ(Wg, ∗), BDiff+(Wg, ∗), BDiffθ(Wg), and BDiff+(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To formulate it we first observe that when π : E → X is a smooth oriented Wg-bundle and H is the local coefficient system over X given by the fibrewise nth homology of this bundle, the fibrewise intersection form λ : H⊗H → Q and its dual ω : Q → H ⊗ H are (−1)n-symmetric and satisfy λ ◦ ω = (−1)n2g · Id, so provide a Q-linear functor S �→ H⊗S from (s)Br2g to the category of local coefficient systems of Q-modules over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (Strictly speaking our definitions require χ = 2 + (−1)n2g to be invertible, so we omit the case (n, g) = (odd, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=') Composing this with taking cohomology gives a functor H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−) : (s)Br2g −→ Gr(Q-mod) S �−→ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The relations in the various spaces of graphs defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1 were chosen precisely to match the contraction formula of [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='10] (in the case of Graph1) and the modified contraction formula of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='6 (in the other cases), so that assigning to a graph its associated κ- or ¯κ-class provides natural transformations (i) κ : Graph1(−)g → H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−), (ii) κ : Graphθ ∗(−)g → H∗(BDiffθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−), (iii) κ : Graph∗(−)g → H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−), (iv) ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−), (v) ¯κ : Graph(−)g → H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−), of functors (s)Br2g → Gr(Q-mod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For 2n = 2 or 2n ≥ 6 the maps (i)–(v) are isomorphisms in a range of cohomological degrees tending to infinity with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We will first give the proof in cases (i), (ii), (iii), and in case (v) assuming case (iv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' the much more involved case (iv) will be treated afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 (i), (ii), (iii), (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For case (i) observe that Graph1(−)g is naturally isomorphic to the functor G(−, V) from [KRW20b, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1], which is shown there to be isomorphic to the functor P(−, V)≥0 ⊗ det⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This case then follows from [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For case (ii) we first construct the homotopy fibre sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) BDiff(Wg, D2n) −→ BDiffθ(Wg, ∗) −→ BSO(2n)⟨n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The left-hand term may be written as the homotopy quotient of Diff(Wg, ∗) acting on the Stiefel manifold Fr(T∗Wg) given by the space of frames in the tangent space to Wg at the point ∗ ∈ Wg, as this action is transitive and its stabiliser is the subgroup which fixes a point and its tangent space, which is homotopy equivalent to fixing a disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The middle term was defined as the homotopy quotient of Diff(Wg, ∗) acting on Bun(T Wg, θ∗γ2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Evaluation at ∗ ∈ Wg defines a Diff(Wg, ∗)-invariant map ev : Bun(T Wg, θ∗γ2n) −→ BSO(2n)⟨n⟩, which is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If we choose a point x ∈ BSO(2n)⟨n⟩ and a framing ξ : (θ∗γ2n)x ∼ → R2n, then there is a map ξ∗ : ev−1(x) → Fr(T∗Wg) given by sending a bundle map ˆℓ : T Wg → θ∗γ2n whose underlying map sends ∗ to x to the framing ξ ◦ ˆℓx : T∗Wg → (θ∗γ2n)x → R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' One verifies by obstruction theory that ξ∗ : ev−1(x) → Fr(T∗Wg) is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Taking homotopy orbits for Diff(Wg, ∗) then gives the required homotopy fibre sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 14 OSCAR RANDAL-WILLIAMS As H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) is spanned by products of twisted Miller–Morita– Mumford classes κεac with c ∈ V in a stable range by (i), and these classes may be defined on BDiffθ(Wg, ∗), the Serre spectral sequence H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) ⇒ H∗(BDiffθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) for the homotopy fibre sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) collapses in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The result then follows by observing that the analogue of the Serre filtration of Graphθ ∗(−)g, induced by the descending filtration by degrees of H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) = V, has gr(Graphθ ∗(−)g) ∼= V ⊗ Graph1(−)g, because modulo V>0 the formula of (d′) specialises to that of (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The induced map gr(κ) : gr(Graphθ ∗(−)g) −→ gr(H∗(BDiffθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−)) therefore has the form V ⊗ {the map κ in case (i)} so is an isomorphism in a stable range by case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Case (ii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Case (iii) is just like the above, using the homotopy fibre sequence BDiff(Wg, D2n) −→ BDiff+(Wg, ∗) −→ BSO(2n) instead, which is established in the analogous way, and W in place of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Case (v) can be deduced from case (iv) by applying the same method to the homotopy fibre sequence BDiffθ(Wg) −→ BDiff+(Wg) ξ −→ BSO[0, n] established in [GRW19, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The filtration step is a little different, so we give some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It follows from (iv) that H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) is spanned by products of twisted Miller–Morita–Mumford classes κ¯εac with c ∈ V in a stable range, and these may be defined on BDiff+(Wg) (in fact they may be defined even for c ∈ W) so the corresponding Serre spectral sequence H∗(BSO[0, n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) ⇒ H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) degenerates in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In this case the analogue of the Serre filtration on Graph(−)g is induced by giving the graph Υi := ({0}, ∅, ∅ → {0}, ∅, c(0) = epi) filtration 4i for 1 ≤ i ≤ ⌊n/4⌋, giving all other connected graphs filtration 0, and extending multiplicatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The associated graded of this filtration has the form gr(Graph(−)g) ∼= Q[Υ1, Υ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , Υ⌊n/4⌋] ⊗ Graphθ(−)g, because the relation in (c′′′′) shows that any graph with a vertex labelled cpi for 1 ≤ i ≤ ⌊n/4⌋ is equivalent to a graph of strictly larger filtration, unless the vertex is 0-valent and the label is epi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As ¯κ(Υi) = κepi = χ · ξ∗(pi) by [GRW19, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5] it follows that the induced map gr(¯κ) : gr(Graph(−)g) −→ gr(H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S)) has the form {an isomorphism} ⊗ {the map ¯κ in case (iv)} so is an isomorphism in a stable range by case (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 (iv) is of a less formal nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It will be parallel to that of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15], but algebraically more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' An important tool will be the following lemma, inspired by [Qui71, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 566].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let G be a topological group and p : P → X be a principal G-bundle with action a : G×P → P, which satisfies the Leray–Hirsch property in cohomology over a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Then H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' F) H∗(P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' F) H∗(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' F) ⊗F H∗(P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' F) p∗ a∗ 1⊗Id ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 15 is an equaliser diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let us leave F implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By the Leray–Hirsch property H∗(P) is a free H∗(X)-module and hence is faithfully flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus it suffices to prove that the dia- gram is an equaliser diagram after applying −⊗H∗(X) H∗(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By the Leray–Hirsch property we also have H∗(P) ⊗H∗(X) H∗(P) ∼ → H∗(P ×X P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus it suffices to show that H∗(P) H∗(P ×X P) H∗(G) ⊗ H∗(P ×X P) pr∗ 2 a∗ 1⊗Id is an equaliser diagram, which is the same question for the principal G-bundle pr2 : P ×X P → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' But this principal G-bundle has a section given by the diagonal map, which trivialises it: this trivialisation identifies the diagram with H∗(P) H∗(G) ⊗ H∗(P) H∗(G) ⊗ H∗(G) ⊗ H∗(P) 1⊗Id µ∗⊗Id 1⊗Id which is indeed an equaliser diagram as it has a contraction induced by a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ We adapt the proof of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15], supposing for concreteness that n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider the tangential structure θ × Y : BSO(2n)⟨n⟩ × Y → BSO(2n) with Y = K(W ∨, n + 1) and W a generic rational vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Then we have H∗(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ∼= Sym∗(W[n + 1]), the symmetric algebra on the vector space W places in (even) degree n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If n is even then like at the end of the proof of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15] we would take Y = K(W ∨, n + 2) instead, so H∗(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) would still be a symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Apart from this there is no essential difference, and we will not comment further on the differences in the case n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There are associated universal Wg-bundles π : Eθ −→ BDiffθ(Wg) πY : Eθ×Y −→ BDiffθ×Y (Wg) and an evaluation map ℓ : Eθ×Y → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Neglecting the “maps to Y ” part of the tangential structure gives a homotopy fibre sequence map(Wg, Y ) −→ BDiffθ×Y (Wg) −→ BDiffθ(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We can take Y to be a topological abelian group, which then acts fibrewise on the map θ × Y and hence acts on compatibly Eθ×Y and BDiffθ×Y (Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using this we can form the homotopy fibre sequence map(Wg, Y )//Y −→ BDiffθ×Y (Wg)//Y −→ BDiffθ(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The space map(Wg, Y )//Y is a K(Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)⊗W ∨, 1), so there is an identification of graded local coefficient systems H∗(map(Wg, Y )//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) = Λ∗(H ⊗ W[1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This is natural in the vector space W, and scaling by u ∈ Q× acts on Λk(H⊗ W[1]) by uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It follows that it acts this way on the kth row of the Serre spectral sequence Ep,q 2 = Hp(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Λq(H ⊗ W[1])) ⇒ Hp+q(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As the differentials in this spectral sequence must be equivariant for this Q×-action, it follows that they must all be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Furthermore this action gives a weight decomposition of both sides, which identifies H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Λk(H ⊗ W)) ∼= H∗+k(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)(k), the weight k-subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To access the latter groups, we use that there is a map α : BDiffθ×Y (Wg) −→ Ω∞ 0 (MTθ ∧ Y+) 16 OSCAR RANDAL-WILLIAMS which by the main theorems of [Bol12, RW16, GTMW09] (for 2n = 2) and [GRW18, GRW14, GRW17] for (2n ≥ 6) is an isomorphism on cohomology in a stable range of degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Here MTθ is the Thom spectrum of −θ∗γ2n, so writing u−2n ∈ H−2n(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) for its Thom class, by the Thom isomorphism we have H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ∼= u−2n · H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) = u−2n · Q[p⌈n+1 4 ⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , pn−1, e].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The rational cohomology of Ω∞ 0 (MTθ ∧ Y+) is then given by Sym∗([H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ Sym∗(W[n + 1])]>0), which can be considered as the free (graded-)commutative algebra on the even- degree classes κc,w1···wr with c ∈ Q[p⌈ n+1 4 ⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , pn−1, e] and wi ∈ W, modulo lin- earity in c and in the wi, and modulo commutativity of the wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The pullbacks of these classes along α we again denote κc,w1···wr, and they may be described intrinsically as the fibre integrals πY !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (c(TπY Eθ×Y ) · ℓ∗(w1 · · · wr)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There are unique classes ¯κc,w1···wr ∈ H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) which pull back to � I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',r} κc,wI · � j∈J (− 1 χκe,wj) ∈ H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q), and in a stable range of degrees H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is the free graded-commutative algebra on the classes ¯κc,w1···wr, modulo linearity in c and in the wi, commutativity of the wi, and modulo ¯κe,w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We wish to apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5 to the principal Y -bundle (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2) BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg)//Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' First observe that the fibre inclusion j : Y → BDiffθ×Y (Wg) classifies the Wg- bundle pr1 : Y × Wg → Y equipped with the product θ-structure and the map ℓ = pr1 : Y × Wg → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus for any w ∈ W we have j∗κe,w = χw ∈ Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q), and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2) satisfies the Leray–Hirsch property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5 then describes H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) as the equaliser of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3) H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) H∗(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' a∗ 1⊗Id In a stable range H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is described in terms of the classes κc,w1···wr, so to make use of this equaliser description we must determine how these classes pull back along the action map a : Y × BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This map classifies the Wg-bundle Y × πY : Y × Eθ×Y → Y × BDiffθ×Y (Wg) equipped with the structure map Y × Eθ×Y Y ×ℓ → Y × Y → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As the wi ∈ W = Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) are primitive with respect to the coproduct induced by the multiplication on Y , we have a∗(κc,w1···wr) = (Y × πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ((1 × c(TπY Eθ×Y )) · r � i=1 (wi × 1 + 1 × ℓ∗(wi))) = � I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',r} wI × κc,wJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 17 Our goal now is to show that the classes defined by ¯κc,w1···wr := � I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',r} κc,wI · � j∈J (− 1 χκe,wj) ∈ H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) are equalised by the maps (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3), so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5 descend to unique classes of the same name in H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To see this, we calculate using the formula above that a∗(¯κc,w1···wr) = � I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',r} � � S⊔T =I wS × κc,wT � (−1)|J| � j∈J (wj × 1 + 1 χ1 × κe,wj) = � S⊔T ⊔U⊔V ={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',r} (−1)|U|wS⊔U × � κc,wT · � v∈V (− 1 χκe,wv) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For each A ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=', r} the coefficient of wA is \uf8eb \uf8ed � U⊆A (−1)|U| \uf8f6 \uf8f8 \uf8eb \uf8ed � T ⊔V ={1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',r}\\A κc,wT · � v∈V (− 1 χκe,wv) \uf8f6 \uf8f8 and � U⊆A(−1)|U| vanishes if A ̸= ∅, and is 1 if A = ∅ (it is the binomial expansion of (1 − 1)|A|), which shows that a∗(¯κc,w1···wr) = 1 × ¯κc,w1···wr as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Finally, that these classes (except ¯κe,w1 = 0) freely generate the Q-algebra H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) in a stable range follows from the fact that the κc,w1···wr freely generate H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) in a stable range, together with the observa- tion that ¯κc,w1···wr ≡ κc,w1···wr modulo the ideal generated by classes κe,w and the Leray–Hirsch property again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ Let us provide a “fibre-integral” interpretation of the classes we have just con- structed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider the map of principal Y -bundles Y Eθ×Y Eθ×Y //Y Y BDiffθ×Y (Wg) BDiffθ×Y (Wg)//Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' i πY πY //Y j The composition ℓ ◦ i : Y → Y is the identity, so i∗ℓ∗(w) = w ∈ Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We showed in the proof above that j∗κe,w = χw ∈ Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q), so in particular both these principal Y -bundles satisfy the Leray–Hirsch property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Together these give that i∗(ℓ∗(w) − 1 χ(πY )∗κe,w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As Y is n-connected it follows from the Serre spectral sequence that there exists a unique class ¯ℓ∗(w) ∈ Hn+1(Eθ×Y //Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) which pulls back to ℓ∗(w) − 1 χ(πY )∗κe,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We have ¯κc,w1···wr = (πY //Y )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) ∈ H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As the lower of the above principal Y -bundles satisfies the Leray–Hirsch property, this identity may be verified after pulling back to BDiffθ×Y (Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In H∗(Eθ×Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) we have ¯ℓ∗(w) = ℓ∗(w) − 1 χ(πY )∗κe,w, so expanding out gives (πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) = (πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (c · r � i=1 (ℓ∗(wi) − 1 χ(πY )∗κe,wi)) = � I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',r} κc,wI · � j∈J (− 1 χκe,wj) 18 OSCAR RANDAL-WILLIAMS as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ The classes ¯κc,w1···wr provide an isomorphism Sym∗ �[H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ Sym∗(W[n + 1])]>0 u−2n · e ⊗ W[n + 1] � −→ H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) in a stable range, natural in W, which with the discussion above gives an identifi- cation of graded vector spaces H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Λ∗(H ⊗ W[1])) ∼= Sym∗ �[H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ Sym∗(W[n + 1])]>0 u−2n · e ⊗ W[n + 1] � natural in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Just as in the proof of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15], and using its notation, this implies that there is a natural transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4) Pbis(−, V)≥0 ⊗ det⊗n −→ H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−) of lax symmetric monoidal functors FB → Gr(Q-mod) which is an isomorphism in a stable range, where P(−, V)≥0 → Pbis(−, V)≥0 is the quotient by those partitions containing a part of size 1 labelled by e ∈ V2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Assigning to a labelled part the corolla with that label gives a natural transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5) Pbis(−, V)≥0 ⊗ det⊗n −→ Graphθ(−)g, of lax symmetric monoidal functors FB → Gr(Q-mod), and we claim that using this (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4) factors through the map ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Assuming this claim for now, observe that using the contraction relations in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1 (iv) (d′′′) to contract all edges shows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5) is surjective, which with the fact that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4) is an isomorphism in a stable range will show that the map ¯κ is an isomorphism in a stable range too (as well as the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It remains to show the factorisation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' that the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4) sends a part of size a labelled by c ∈ V to the class κ¯εac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We again proceed as in the relevant step of the proof of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There is a fibration sequence map(Wg, Y ) −→ Eθ×Y −→ Eθ and so, taking homotopy orbits for the fibrewise Y -action, a fibration sequence map(Wg, Y )//Y −→ Eθ×Y //Y −→ Eθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Again by functoriality in W the associated Serre spectral sequence collapses to identify the weight decomposition as H∗(Eθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Λk(H ⊗ W)) ∼= H∗+k(Eθ×Y //Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Given the description in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7 we must show that the map ¯ℓ(−) : W −→ Hn+1(Eθ×Y //Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)(1) ∼= Hn(Eθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H) ⊗ W is given by w �→ ¯ε ⊗ w, which is the analogue of [KRW20b, Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As it is natural in the vector space W it must certainly be given by ¯ℓ(w) = x ⊗ w for some x ∈ Hn(Eθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H), and we must show that x = ¯ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' That the restriction of x to the fibre Wg of π : Eθ → BDiffθ(Wg) is given by coevaluation may be done precisely as in [KRW20b, Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By the characterisation of ¯ε it remains to check that 1 χ(πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (e · ¯ℓ∗(w)) = 0 ∈ Hn+1(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By the Leray–Hirsch property this may be checked after pulling back to BDiffθ×Y (Wg), but as ¯ℓ∗(w) = ℓ∗(w) − 1 χ(πY )∗κe,w ∈ Hn+1(Eθ×Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) by definition, the vanishing is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There are natural maps BDiff(Wg, D2n) BDiffθ(Wg, ∗) BDiffθ(Wg) BDiff(Wg, D2n) BDiff+(Wg, ∗) BDiff+(Wg) a b c d e f which each induce maps on H∗(−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There are corresponding maps of spaces of graphs Graph1(−)g Graphθ ∗(−)g Graphθ(−)g Graph1(−)g Graph∗(−)g Graph(−)g a∗ b∗ e∗ c∗ d∗ f ∗ given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The maps c∗ and d∗ are induced by the projections W → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The maps a∗ and e∗ are induced by applying the augmentations V → Q and W → Q to the second tensor factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The maps b∗ and f ∗ are more subtle, as they involve converting between blue graphs and red graphs, via the formula of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Graphically it is given by �→ − 1 χ( e e e + + ) with certain orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The maps b and f are also oriented Wg-bundles, so they also induce fibre- integration maps b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' and f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' on cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' These are b∗- and f ∗-linear respectively, so are determined by the maps (of degree −2n) b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' : V −→ Graphθ(−)g f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' : W −→ Graph(−)g which each send a monomial c in pi’s and e to the graph given by a single vertex labelled by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Cohomology of Torelli groups The isomorphisms provided by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 can be converted into information about the spaces BTor(Wg, D2n), BTorθ(Wg, ∗), BTor+(Wg, ∗), BTorθ(Wg), BTor+(Wg) just as [KRW20a, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1] is deduced from [KRW20a, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let us give the definition of these spaces and formulate the result: the following is largely a reminder of some points from [KRW20a], and we do not spell out all details again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The group Diff+(Wg) acts on Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Z) preserving the nondegenerate (−1)n- symmetric intersection form λ : Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Z) ⊗ Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Z) → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This provides a homomorphism αg : Diff+(Wg) −→ Gg := � Sp2g(Z) if n is odd, Og,g(Z) if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This map is not always surjective, but its image is a certain finite-index subgroup G′ g ≤ Gg, an arithmetic group associated to the algebraic group Sp2g or Og,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This subgroup has been determined by Kreck [Kre79]: it is the whole of Gg if n is even or n = 1, 3, 7, and otherwise is the subgroup Spq 2g(Z) ≤ Sp2g(Z) of those matrices which preserve the standard quadratic refinement (of Arf invariant 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 20 OSCAR RANDAL-WILLIAMS We define Tor+(Wg) to be the kernel of this homomorphism, and Tor+(Wg, ∗) and Tor(Wg, D2n) to be the kernel of its restriction to the subgroups Diff+(Wg, ∗) and Diff(Wg, D2n) respectively (these restrictions still have image G′ g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Further- more, we define BTorθ(Wg) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg) BTorθ(Wg, ∗) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg, ∗), where Bun+(T Wg, θ∗γ2n) ⊂ Bun(T Wg, θ∗γ2n) consists of the orientation-preserving bundle maps (for some choice of orientation of θ∗γ2n that we make once and for all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By the discussion at the beginning of Section 3 the spaces Bun+(T Wg, θ∗γ2n) are path-connected, so each of the BTor’s we have defined are principal G′ g-bundles over the corresponding BDiff’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In particular, their rational cohomologies are both Q-algebras and G′ g-representations, and we will describe them as such in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Before doing so, we recall that the work of Borel identifies H∗(G′ g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) = � Q[σ2, σ6, σ10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='] if n is odd, Q[σ4, σ8, σ12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='] if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' in a stable range of degrees, where σ4i−2n may be chosen so that it pulls back to the Miller–Morita–Mumford class κLi ∈ H4i−2n(BDiff+(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) associated to the ith Hirzebruch L-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In particular the κLi vanish in the cohomology of BTor+(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let us write H(g) := Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q), which is the standard representation of G′ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Pulled back from BDiff+(Wg) to BTor+(Wg) the coefficient system H is canonically trivialised, but has an action of G′ g: it can be identified with the dual H(g)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The edge homomorphism of the Serre spectral sequence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) −→ � H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ (H(g)∨)⊗S�G′ g allows us to consider the modified twisted Miller–Morita–Mumford classes ¯κεSc as providing G′ g-equivariant homomorphisms ¯κc : H(g)⊗S −→ Hn(|S|−2)+|c|(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The identities from the modified contraction formula correspond to identities among these maps: this will give relations analogous to [KRW20b, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2], which we will spell out after the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' First we explain how these relations can be organised in a categorical way, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Considering (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) as a natural transformation of functors on (s)Br2g, we may precompose it with the map ¯κ : Graphg(−) −→ H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−) (which is an isomorphism in a stable range for n ̸= 2 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This gives G′ g-equivariant maps H(g)⊗S ⊗ Graphg(S) → H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) which assemble to a map K∨ ⊗(s)Br2g Graphg(−) −→ H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) out of the coend, where K : (s)Br2g → Rep(G′ g) sends S to H(g)⊗S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The domain obtains a graded-commutative Q-algebra structure coming from the lax symmetric monoidality of Graphg(−) and strong symmetric monoidality of K(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1 below will say that this is surjective in a stable range, with kernel the ideal generated by the κLi, but before stating it we explain a simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let us write i : d(s)Br → (s)Br2g for the inclusion of the downward (signed) Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus subcategory is independent if g, as no circles can be created by composing morphisms in the downward Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Write Graph1(−)′ ⊂ i∗Graph1(−)g for the subfunctor where we forbid bivalent vertices labelled by 1 ∈ V both of whose half-edges are legs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' similarly, this functor is independent of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Like ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 21 just after [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='11], Graph1(−)g is then the left Kan extension i∗Graph1(−)′ of Graph1(−)′ along i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We similarly define Graphθ ∗(−)′, Graph∗(−)′, Graphθ(−)′, and Graph(−)′, whose left Kan extensions again recover the original functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The following is the analogue of [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There are G′ g-equivariant ring homomorphisms i∗(K∨) ⊗d(s)Br Graph1(−)′ (κLi | 4i − 2n > 0) −→ H∗(BTor(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) (i) i∗(K∨) ⊗d(s)Br Graphθ ∗(−)′ (κLi | 4i − 2n > 0) −→ H∗(BTorθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) (ii) i∗(K∨) ⊗d(s)Br Graph∗(−)′ (κLi | 4i − 2n > 0) −→ H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) (iii) i∗(K∨) ⊗d(s)Br Graphθ(−)′ (κLi | 4i − 2n > 0) −→ H∗(BTorθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) (iv) i∗(K∨) ⊗d(s)Br Graph(−)′ (κLi | 4i − 2n > 0) −→ H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) (v) which for 2n ≥ 6 are isomorphisms in a stable range of degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If 2n = 2 then, in a stable range of degrees and assuming that the target is finite-dimensional in degrees ∗ < N for all large enough g, these maps are iso- morphisms onto the maximal algebraic subrepresentations in degrees ∗ ≤ N, and monomorphisms in degrees ∗ ≤ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By the main theorem of [KRW20a], as long as 2n ≥ 6 the G′ g-representations Hi(BTor(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) are algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using the inheritance properties for algebraic representations from [KRW20a, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2], the Serre spectral sequences for the homotopy fibre sequences BTor(Wg, D2n) −→BTor+(Wg, ∗) −→ BSO(2n) BTor(Wg, D2n) −→BTorθ(Wg, ∗) −→ BSO(2n)⟨n⟩ show that the cohomology groups of BTor+(Wg, ∗) and BTorθ(Wg, ∗) are also al- gebraic G′ g-representations, and the same for the homotopy fibre sequences Wg −→BTor+(Wg, ∗) −→ BTor+(Wg) Wg −→BTorθ(Wg, ∗) −→ BTorθ(Wg) show that the cohomology groups of BTor+(Wg) and BTorθ(Wg) are algebraic G′ g-representations too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using this algebraicity property, case (i) is precisely [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1], using that by [KRW20b, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1] Graph1(−)g is isomorphic to the functor P(−, V)≥0⊗det⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The other cases follow in the same way, using [KRW20b, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='16], from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4, with one elaboration which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The addendum in the case 2n = 2 is precisely as in [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The elaboration comes when verifying the first hypothesis of [KRW20b, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3], which in case (v) for example requires us to know that H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) is a free H∗(G′ g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)-module in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' But by transfer H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) is a summand of H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) (as H∗(G′ g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)-modules), and similarly with θ-structures, so cases (ii) and (iii) imply cases (iv) and (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In the other hand in case (iii) for example we have discussed in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 the degeneration of the Serre spectral sequence in a stable range, giving gr(H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S)) ∼= H∗(BSO(2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) ⊗ H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 22 OSCAR RANDAL-WILLIAMS The Serre filtration is one of H∗(G′ g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)-modules, so as the associated graded is a free H∗(G′ g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q)-module in a stable range (because H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) is the case treated in [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1]), it follows that H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗S) is too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The same argument applies in case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ This quite categorical description can be used to get a more down-to-earth pre- sentation for these cohomology rings: in case (v) this is the presentation we have recorded in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This is deduced just as in [KRW20b, Section 5], though most of the work has been done as we have already expressed things in terms of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As in [KRW20b, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4] this is not the smallest possible presentation: it can be simplified by manipulating graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' we leave the details to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The case 2n = 2 Although Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1 is only known to hold in a limited range of degrees in the case 2n = 2 (N = 2 is currently the best known constant for g ≥ 3, using the work of Johnson [Joh85]), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 does hold in a range of cohomological degrees tending to infinity with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In this case our discussion is closely related to the work of Kawazumi and Morita [Mor96, KM96, KM01], and in this section we we take the opportunity to revisit that work from our perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Throughout this section we assume that g ≥ 2, so that χ(Wg) = 2 − 2g ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In terms of Kawazumi and Morita’s notation we have Mg := π0(Diff+(Wg)) Mg,∗ := π0(Diff+(Wg, ∗)) Mg,1 := π0(Diff+(Wg, D2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Under our assumption g ≥ 2 the groups Diff+(Wg), Diff+(Wg, ∗), and Diff+(Wg, D2) all have contractible path-components, so the group cohomology of Mg is the co- homology of BDiff+(Wg), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 gives a natural transformation ¯κ : Graph(−)g −→ H∗(Mg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' H⊗−) of functors sBr2g → Gr(Q-mod), which is an isomorphism in a stable range of degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Note that in this case H∗(BSO(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) = Q[e] so V = W = Q[e] and there is no difference between the tangential structure θ and an orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In particular if we denote by Γi ∈ Graph(∅) the graph with a single vertex, no edges, and labelled by ei+1, then ¯κ(Γi) = κi ∈ H2i(Mg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Q) is the usual Miller–Morita–Mumford class4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Our goal in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 is to analyse Graph(−) in several ways, making contact with the work of Kawazumi and Morita mentioned above as well as work of Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Reduction to corollas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The possible labels for the vertices of graphs in Graph(S) are powers of the Euler class e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Given any graph we may iteratedly apply the modified contraction formula to write it as a linear combination of graphs with fewer edges, and hence any graph is equivalent to a linear combination of graphs with no edges: these are disjoint unions of corollas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Of these, by definition of Graph: the 0-valent corolla labelled by e is equal to the scalar χ, the 1-valent corolla labelled by 1 ∈ V is trivial, and the 1-valent corolla labelled by e ∈ V is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Define a labelled partition of a finite set S to be a partition {Sα}α∈I of S into (possibly empty) subsets and a label enα for each part, such that (i) If |Sα| = 0 then nα ≥ 2, (ii) If |Sα| = 1 then nα ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 4Our κi is denoted ei in the work of Kawazumi and Morita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 23 We give a part (Sα, nα) degree 2nα + |Sα| − 2, and a labelled partition the degree given by the sums of the degrees of its parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Similarly to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 (iv) (particularly around equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5)), let Pbis(S, V)≥0 denote the free Q[χ±1]- module with basis the set of labelled partitions of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Assigning to a labelled part (Sα, enα) the corolla with legs Sα and label enα defines a map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) Pbis(S, V)≥0 ⊗ det QS −→ Graph(S), natural in S with respect to bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It is surjective, as explained above, by repeatedly applying the modified contraction formula to express a graph in terms of graphs without edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If it were not injective then it would have some nontrivial Q[χ±1]-linear com- bination of labelled partitions in its kernel, of a given degree d, and this would remain a nontrivial Q-linear combination of labelled partitions when specialised to χ = 2 − 2g for all g ≫ 0 (as a Laurent polynomial in χ has finitely-many roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' But in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 (iv), in the discussion after equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5), it is explained that when specialised to χ = 2 − 2g this map is an isomorphism in a range of degrees tending to infinity with g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' for large enough g the degree d will be in this stable range, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ In particular, for the graphs Γi described above there is an isomorphism (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2) Q[χ±1][Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='] ∼= Graph(∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Reduction to trivalent graphs without labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In this section we will prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using the modified contraction formula any marked oriented graph is equivalent to a Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1]-linear combination of trivalent graphs with all vertices labelled by 1 ∈ V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let Graphtri(S) ≤ Graph(S) denote the sub-Q[χ±1]-module spanned by those marked oriented graphs which are trivalent and all of whose labels are 1 ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The monomorphism i : Graphtri(−) → Graph(−) becomes an iso- morphism upon inverting χ − 2, χ − 3, and χ − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In particular Graphtri(−)g = Graph(−)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 (2-valent vertices labelled by 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using the relation λ2,3(κ¯ε1,2κ¯ε3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',nc) = κ¯ε1,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=',nc we can always remove 2-valent vertices labelled by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It is sometimes convenient when writing formulas for 3-valent graphs to also allow 2-valent vertices labelled by 1: we allow ourselves to do so, noting that the above can always be used to eliminate the 2-valent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As a matter of notation we will formally manipulate modi- fied twisted Miller–Morita–Mumford classes, but this is equivalent to manipulating marked oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Rearranging the first contraction formula gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3) κ¯εaeb = χ χ−2 � λ1,2κ¯ε2+aeb−1 − 1 χ2 κe2κ¯εaeb−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Rearranging the second contraction formula gives κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) − 1 χ2 (κe2 · κ¯εa · κ¯εb) + 1 χ(κ¯εae · κ¯εb + κ¯εa · κ¯εbe) 24 OSCAR RANDAL-WILLIAMS and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3) to eliminate the Euler classes from the last two terms gives κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) − 1 χ2 (κe2 · κ¯εa · κ¯εb) + 1 χ−2((λ1,2(κ¯ε2+a) − 1 χ2 κe2κ¯εa) · κ¯εb + κ¯εa · (λ1,2(κ¯ε2+b) − 1 χ2 κe2κ¯εb)) = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) + 1 χ−2 ((λ1,2(κ¯ε2+a) · κ¯εb + κ¯εa · λ1,2(κ¯ε2+b)) − 1 χ(χ−2)κe2 · κ¯εa · κ¯εb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It suffices to show that each corolla κ¯εaeb may be represented by a linear combi- nation of trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='8 the class κe2 may be represented by a trivalent graph (after inverting χ − 3) so by iteratedly applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3) it suffices to show that each κ¯εn can too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' By Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4 we may as well show that classes can be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' To get started we have κ¯ε = 0 as it has negative degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider the class λ2,5λ3,4(κ¯ε1,2,3 ·κ¯ε4,5,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using the form of the relations above, which avoid creating Euler classes, this is λ2,5(κ¯ε1,2,5,6 − 1 χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6 + κ¯ε1,2λu,v(κ¯εu,v,5,6)) + 1 χ(χ−2)(κe2κ¯ε1,2κ¯ε5,6)) = λ2,5(κ¯ε1,2,5,6) − 2 χ−2λu,v(κ¯εu,v,1,6) + 1 χ(χ−2)κe2κ¯ε1,6 = χ−4 χ−2λ2,5(κ¯ε1,2,5,6) + 1 χ(χ−2)κe2κ¯ε1,6 Renumbering legs and rearranging, this shows that λ1,2(κ¯ε4) may be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Applied with (a, b) = (2, 2) the second relation gives κ¯ε4 = λ3,4(κ¯ε3 · κ¯ε3) + 1 χ−2 ((λ1,2(κ¯ε4) · κ¯ε2 + κ¯ε2 · λ1,2(κ¯ε4)) − 1 χ(χ−2)κe2 · κ¯ε2 · κ¯ε2, which with the above shows that κ¯ε4 may be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Similarly to the above, consider λ2,5λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6,7), which is λ2,5(κ¯ε1,2,5,6,7 + 1 χ(χ−2)κe2κ¯ε1,2κ¯ε5,6,7 − 1 χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6,7 + κ¯ε1,2λu,v(κ¯εu,v,5,6,7))) = λ2,5(κ¯ε1,2,5,6,7) + 1 χ(χ−2)κe2κ¯ε1,6,7 − 1 χ−2(λ2,5λu,v(κ¯εu,v,1,2κ¯ε5,6,7) + λu,v(κ¯εu,v,1,6,7)) = χ−3 χ−2λ2,5(κ¯ε1,2,5,6,7) + 1 χ(χ−2)κe2κ¯ε1,6,7 − 1 χ−2λ2,5λu,v(κ¯εu,v,1,2)κ¯ε5,6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Renumbering legs and rearranging, this shows that λ1,2(κ¯ε5) may be represented by 2-, 3-, and 4-valent graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' with the above it follows that it can also be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Applied with (a, b) = (2, 3) the second relation gives κ¯ε5 = λ3,4(κ¯ε3 · κ¯ε4) + 1 χ−2 ((λ1,2(κ¯ε4) · κ¯ε3 + κ¯ε2 · λ1,2(κ¯ε5)) − 1 χ(χ−2)κe2 · κ¯ε2 · κ¯ε3, so it follows that κ¯ε5 may be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If n ≥ 6 then we can write n = a + b with a, b ≥ 3, so a + 2, b + 2 < n and so the second relation expresses κ¯εn in terms of κ¯εm’s with m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus all κ¯εn’s may be represented by 2- and 3-valent graphs as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ It is worth observing that we have the relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4) λ1,2(κ¯ε3) = χ−2 χ κ¯εe + 1 χ2 κe2κ¯ε = 0, using that κ¯εe = 0 (by definition) and that κ¯ε = 0 (as it has negative degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This means that any graph having a trivalent vertex with a loop is trivial in Graph(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' A remark on orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' A curious normalisation is possible when consider- ing trivalent graphs, allowing one to neglect the orderings of vertices, of half-edges, and the orientations of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In [Mor96, KM96, KM01] this is implemented ab initio and (marked) oriented graphs play no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let us explain this normalisation, extended to trivalent graphs with legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' A trivalent graph ˜Γ with legs S consists of a set V of vertices, a set H of half- edges, a 3-to-1 map a : H → V recording to which vertex each half-edge is incident, and an unordered matching µ on H ⊔ S recording which half-edges span an edge, and which half-edges are connected to which legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Given a trivalent graph ˜Γ = (V, H, a : H → V, µ) with legs S, we may choose an ordering of V and choose an ordering of H such that a : H → V is weakly monotone (equivalently, choose an ordering of the half-edges incident at each vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We also choose an ordering of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' There is an induced ordering of H ⊔ S by putting ⃗S after ⃗H, and we form an ordered matching m of H ⊔ S by taking those pairs (a, b) with a < b and {a, b} ∈ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using this we form an oriented trivalent graph Γchoice = (⃗V , ⃗H, a : H → V, m), depending on these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The normalisation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let x1 < x2 < x3 < x4 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' < x2k ∈ H ⊔ S be the total order on H ⊔ S, and let a1 < b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' , ak < bk be the ordered pairs which span an edge, with a1 < a2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' < ak ∈ H ⊔ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Then there is a bijection given by ρ := � a1 b1 a2 b2 a3 b3 ··· ak bk x1 x2 x3 x4 x5 x6 ··· x2k−1 x2k � and we define Γ := sign(ρ) · Γchoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' As long as ˜Γ has no vertices with loops, the element Γ does not depend on the choice of ordering of V or H, and depends on the ordering of S precisely as the sign representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In particular if we set5 Graphundec(S) := Q[χ±1][˜Γ trivalent graph with legs S]/(graphs with loops) then the Claim together with the relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4) provides an epimorphism Φ : Graphundec(S) ⊗ det QS −→ Graphtri(S) of Q[χ±1]-modules, natural with respect to bijections in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This can be extended to a natural transformation of functors on sBrχ by letting an ordered matching (a, b) of elements of S act by adding an edge to the trivalent graph connecting a and b, and contracting the determinant by a ∧ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Doing so might create a circle with no vertices, which should be replaced by the scalar χ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If (h1, h2, h3) are the half-edges incident at a vertex v and we change their ordering to (hσ(1), hσ(2), hσ(3)) giving Γ′ choice, then (under the assump- tion that Γ does not have loops) the relative ordering of half-edges forming an edge has not changed, so m′ = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus Γ′ choice = sign(σ) · Γchoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' On the other hand ρ′ is obtained from ρ by postcomposing with σ, and precomposing with a permutation which permutes some (ai < bi)’s, which is an even permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus sign(ρ′) = sign(σ) · sign(ρ), so Γ′ = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Suppose a vertex v1 has half edges (h1 1, h1 2, h1 3) and v2 has half edges (h2 1, h2 2, h2 3), and v1 < v2 ∈ ⃗V are adjacent in the ordering on V , and consider transposing the ordering of these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For edges between a u < v1 and a vi or between a vi and a u > v2 the relative ordering of their half-edges does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Edges between v1 and v2 have the relative ordering of their half-edges reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus if there are N such edges we have Γ′ choice = (−1)1+N ·Γchoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' But the permutation ρ is changed 5In [Mor96, KM96, KM01] they restrict to “trivalent graphs without loops”, however we find it more natural to allow loops but set graphs with a loop to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 26 OSCAR RANDAL-WILLIAMS by permuting (h2 1, h2 2, h2 3) past (h1 1, h1 2, h1 3), which has sign −1, and N transpositions (aibi), which has sign (−1)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus again Γ′ = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Finally, changing the order of S by a permutation τ changes ρ by postcomposition with τ, so acts as sign(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For the ordering of vertices and half-edges corresponding to the theta-graph in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='8 the associated permutation is ρ = (1)(235)(46) which is odd, so the undecorated theta-graph yields χ−3 χ κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This is precisely minus the evaluation of βΓ2 on [KM01, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 39] (unfortunately the theta-graph is denoted Γ2 in that paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This minus comes from the use of a different sign convention, see the discussion at [KRW20b, top of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Relations among trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The modified contraction formula de- scribes relations among graphs involving contracting an edge, but this necessarily involves graphs with vertices of different valencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2 we have ex- plained that, in the case of surfaces, all graphs may be expressed purely in terms of trivalent graphs: one may ask what relations among trivalent graphs Γ are imposed by the contraction formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' For the unmodified contraction formula discussed in [KRW20b], the answer is that it imposes the “I = H” relation among trivalent graphs: this is because both the I- and H-graphs admit contractions to the X-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Furthermore, as all connected trivalent graphs with the same number of legs and of the same genus are equivalent under the “I = H” relation, and the contraction formula never changes the genus or number of legs, there are no further relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In the setting of the modified contraction formula discussed here it is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It is best given in the setting of undecorated trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' After inverting χ−2, χ−3, and χ−4, undecorated trivalent graphs which differ locally by (IHmod) = + 1 (χ−4)(3−χ)( − ) + 1 χ−4( + − − ) give the same elements in Graphtri[(χ − 2)−1, (χ − 3)−1, (χ − 4)−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We establish this relation in Graphtri({a, b, c, d})⊗detQ{a,b,c,d}, and it then follows in general using functoriality on the signed Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We order the legs as a < b < c < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (i) a b c d 1 3 5 6 2 4 (ii) a b c d 1 2 6 3 4 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Some marked graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider first the H-shaped graph shown in Figure 3 (i), with the depicted names of half edges, ordered as 3 < 1 < 5 < 6 < 2 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Its corresponding permutation is � 3 c 1 a 5 6 2 b 4 d 3 1 5 6 2 4 a b c d � which is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus this ordering data represents the underlying ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 27 undecorated H-shaped trivalent graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Ignoring for now the matchings to the legs (which are given by matching 1 with a, 2 with b, and so on), it corresponds to λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using the form of the relations which avoid creating Euler classes from the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2 we have λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4) = κ¯ε3,1,2,4 + 1 χ(χ−2)κe2κ¯ε3,1κ¯ε2,4 − 1 χ−2(λu,v(κ¯εu,v,3,1)κ¯ε2,4 + κ¯ε3,1λu,v(κ¯εu,v,2,4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Consider now the I-shaped graph shown in Figure 3 (ii), with the depicted names of the half-edges, ordered as 4 < 3 < 5 < 6 < 1 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Its corresponding permutation is � 4 d 3 c 5 6 1 a 2 b 4 3 5 6 1 2 a b c d � which is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus this ordering data represents minus the underlying undecorated I-shaped trivalent graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Ignoring again the matchings to the legs, it corresponds to λ5,6(κ¯ε4,3,5 · κ¯ε6,1,2) = κ¯ε4,3,1,2 + 1 χ(χ−2)κe2κ¯ε4,3κ¯ε1,2 − 1 χ−2(λu,v(κ¯εu,v,4,3)κ¯ε1,2 + κ¯ε4,3λu,v(κ¯εu,v,1,2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The sum of these two expressions therefore represents the image under Φ of the difference H − I of the underlying undecorated trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Furthermore, κ¯ε4,3,1,2 = −κ¯ε3,1,2,4 so these terms cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' From the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2 we have the identity λu,v(κ¯εu,v,s,t) = χ−2 χ−4λi,jλk,l(κ¯εs,i,k · κ¯εl,j,t) − 1 χ(χ−4)κe2κ¯εs,t, expressing terms of the form λu,v(κ¯εu,v,s,t) in terms of (2- and) 3-valent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Applying it to the sum of the two expressions above, and collecting terms, therefore gives Φ(H − I) = 1 χ(χ−4)κe2� κ¯ε3,1κ¯ε2,4 + κ¯ε4,3κ¯ε2,1� − 1 χ−4 � λi,jλk,l(κ¯ε3,i,k · κ¯εl,j,1)κ¯ε2,4 + κ¯ε3,1λi,jλk,l(κ¯ε2,i,k · κ¯εl,j,4) λi,jλk,l(κ¯ε4,i,k · κ¯εl,j,3)κ¯ε1,2 + κ¯ε4,3λi,jλk,l(κ¯ε1,i,k · κ¯εl,j,2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Using that κe2 = Φ( χ χ−3Θ) and carefully putting the graphs corresponding to the other terms into the normal form of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 gives the identity in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ Our relation IHmod is graphically identical to the relation called IHbis 0 by Akazawa [Aka05, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 100] and in the corrigendum [GN07] to the paper of Garo- ufalidis and Nakamura [GN98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' In those papers it is emphasised that IHbis 0 means this identity is imposed only when the 4 half-edges belong to distinct edges, but in fact this is redundant: if the 4 half-edges do not belong to distinct edges, then the identity already holds in Graphundec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' So in fact imposing our relation IHmod is identical to imposing their relation IHbis 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Upon inverting χ − 2, χ − 3, and χ − 4, the maps Graphundec(S) (IHmod) ⊗ det QS Φ −→ Graphtri(S) inc −→ Graph(S) are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Let R := Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1] and implicitly base change to this ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' We have already shown in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 that the second map is an isomorphism, and Φ is certainly an epimorphism, so it remains to show that the composition is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 28 OSCAR RANDAL-WILLIAMS For an undecorated trivalent graph Γ, define a double edge to be an unordered pair of vertices which share precisely two edges, and a triple edge to be an unordered pair of vertices which share precisely three edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' form a theta-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Define µ(Γ) := 2 · #double edges of Γ + 3 · #triple edges of Γ, filter Graphundec by letting F kGraphundec be spanned by those Γ with µ(Γ) ≥ k, and give Graphundec/(IHmod) the induced filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' If Γ = ΓH is a graph with µ(Γ) = k and a distinguished “H” subgraph, and ΓI is obtained by replacing this “H”-subgraph by “I”, then by applying the relation IHmod to this subgraph we find that (i) if the edge involved is not part of a double or triple edge then the relation gives ΓH − ΓI ∈ F k+1Graphundec/(IHmod), (ii) if the edge involved is part of a double or triple edge then the relation is trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' already holds in Graphundec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus the associated graded of the induced filtration on Graphundec/(IHmod) can be described as Graphundec/(IH0), where as in [GN98] the relation IH0 means imposing the “I = H” relation when the four half-edges belong to different edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Now IH0 is an equivalence relation on the set of isomorphism classes of trivalent graphs without loops, and similarly to [GN98, Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 (c)] it is easy to see that all connected trivalent graph without loops of the same rank and with the same legs are equivalent to each other: in other words the equivalences classes of such are given by partitions of S (the parts are the legs of each connected component) labelled by a power of e (recording the rank of the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' It follows that the rank of Graphundec/(IHmod) in each degree, as an R-module, is at most that of Graph(∅) as determined in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1, and so the composition in the statement of the theorem, which is an epimorphism, must be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' On the work of Garoufalidis and Nakamura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The discussion of the last few sections can be used to complete the work of Garoufalidis and Nakamura [GN98, GN07], concerning the calculation of the invariants [Λ∗V13/(V22)]Sp in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Here we write Vλ for the irreducible Sp-representation corresponding to the partition λ, which was written as [λ]sp in those papers, and V22 denotes the unique copy of this irreducible in Λ2V13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Combining Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 (c) of [GN98] was supposed to calculate [Λ∗V13/(V22)]Sp in a stable range, but for the corrected version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='1 in [GN07], which expresses these invariants as Graphundec(∅)g/(IHbis 0 ), the authors say “it turns out that a simple stable structure of [these invariants] as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 (c) will not be easy to detect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' However Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='7 and equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='2) gives that [Λ∗V13/(V22)]Sp ∼= Graphundec(∅)g/(IHbis 0 ) ∼= Graph(∅)g ∼= Q[Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='] in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Thus in fact Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='3 (c) of [GN98] is correct as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' This can also be obtained from the work of Felder, Naef, and Willwacher [FNW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Specifically, the graded-commutative algebra A(g) defined just before Theorem 6 of that paper is Λ∗V13/(V22), and Theorem 6 together with Proposition 36 (3) also gives the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' References [Aka05] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Akazawa, Symplectic invariants arising from a Grassmann quotient and trivalent graphs, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Okayama Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 47 (2005), 99–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [Bol12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Boldsen, Improved homological stability for the mapping class group with inte- gral or twisted coefficients, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 270 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1-2, 297–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [FNW21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Felder, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Naef, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Willwacher, Stable cohomology of graph complexes, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='12826, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II 29 [GN98] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Garoufalidis and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Nakamura, Some IHX-type relations on trivalent graphs and symplectic representation theory, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 5 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 3, 391–402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [GN07] , Corrigendum: “Some IHX-type relations on trivalent graphs and symplectic representation theory” [Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 5 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 3, 391–402], Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 14 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 4, 689–690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [GRW14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Galatius and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Randal-Williams, Stable moduli spaces of high-dimensional man- ifolds, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 212 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 2, 257–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [GRW17] , Homological stability for moduli spaces of high dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' II, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (2) 186 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1, 127–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [GRW18] , Homological stability for moduli spaces of high dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' I, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 31 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1, 215–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [GRW19] , Moduli spaces of manifolds: a user’s guide, Handbook of homotopy theory, Chapman & Hall/CRC, CRC Press, Boca Raton, FL, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 445–487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [GTMW09] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Galatius, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Tillmann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Madsen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Weiss, The homotopy type of the cobor- dism category, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 202 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 2, 195–239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [Hai20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Hain, Johnson homomorphisms, EMS Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 7 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1, 33–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [Joh85] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Johnson, The structure of the Torelli group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' The abelianization of T , Topology 24 (1985), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 2, 127–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [KM96] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Kawazumi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Morita, The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 3 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 5, 629–641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [KM01] , The primary approximation to the cohomology of the mod- uli space of curves and cocycles for the Mumford-Morita-Miller classes, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='jp/preprint/pdf/2001-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='pdf, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [Kre79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Kreck, Isotopy classes of diffeomorphisms of (k − 1)-connected almost- parallelizable 2k-manifolds, Algebraic topology, Aarhus 1978 (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=', Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Aarhus, Aarhus, 1978), Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 763, Springer, Berlin, 1979, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 643–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [KRW20a] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Kupers and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Randal-Williams, The cohomology of Torelli groups is algebraic, Forum of Mathematics, Sigma 8 (2020), e64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [KRW20b] , On the cohomology of Torelli groups, Forum of Mathematics, Pi 8 (2020), e7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [KRW21] , On the Torelli Lie algebra, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='16010, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [Mor96] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Morita, A linear representation of the mapping class group of orientable sur- faces and characteristic classes of surface bundles, Topology and Teichm¨uller spaces (Katinkulta, 1995), World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=', River Edge, NJ, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 159–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [Qui71] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Quillen, The spectrum of an equivariant cohomology ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' I, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (2) 94 (1971), 549–572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' [RW16] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Randal-Williams, Resolutions of moduli spaces and homological stability, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' (JEMS) 18 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' 1, 1–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content=' Email address: o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='randal-williams@dpmms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'} +page_content='uk Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}