diff --git "a/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/load_file.txt" "b/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/BtE1T4oBgHgl3EQfVwQu/content/tmp_files/load_file.txt" @@ -0,0 +1,626 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf,len=625 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='03105v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='GT] 8 Jan 2023 FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES NIMA ANVARI AND IAN HAMBLETON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Given a 4-manifold with a homologically trivial and locally-linear cyclic group action, we obtain necessary and sufficient conditions for the existence of equi- variant bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The conditions are derived from the twisted signature formula and are in the form of congruence relations between the fixed point data and the isotropy representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Introduction Finite group actions on 4-manifolds can be studied in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We are mainly interested in comparing smooth actions with those which are topological and locally linear, but important examples arise for symplectic 4-manifolds and complex surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Here is a sampling of survey articles and recent work on aspects of this general theme: [1, 2, 3, 5, 6, 7, 10, 15, 22, 20, 21, 23, 26, 31, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We will focus on the existence and classification of equivariant bundles, and their applications in Yang-Mills gauge theory to the study of finite group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We begin by recalling some standard definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) denote a simply-connected, closed 4-manifold X together with a locally linear and homologically trivial action of a cyclic group π = Z/p of prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The fixed point set Xπ has Euler characteristic χ(Xπ) = b2(X) + 2, by the Lefschetz fixed point formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In general Xπ will consist of isolated fixed points and a disjoint union of fixed 2-spheres (see [9, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' At each isolated fixed point x ∈ Xπ, the tangent space admits an equi- variant decomposition (TxiX, π) = C(ai) ⊕ C(bi) of complex representation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let t · (z1, z2) = (ζaiz1, ζbiz2) denote the action for t a fixed generator in the cyclic group π, ζ = e2πi/p, and with integers (ai, bi), both non-zero modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (i) The integers (ai, bi) are the local tangential rotation data, and are well-defined up to order and simultaneous change in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (ii) Similarly, for each point x on a π-fixed 2-sphere Fj, there is a representation C ⊕ C(cj) corresponding to the equivariant splitting TX | Fj = TFj ⊕ N(Fj) where N(Fj) is the normal bundle with rotation ζcj, and cj ̸≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Date: January 4, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This research was partially supported by NSERC Discovery Grant A4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1 2 NIMA ANVARI AND IAN HAMBLETON (iii) The total fixed point rotation data is the collection F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' where I and J index the isolated fixed points and 2-spheres respectively and αj = [Fj]·[Fj] is the self-intersection number of the fixed spheres [Fj] ∈ H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (iv) By an equivariant line bundle (L, π) → (X, π) we mean a principal U(1)-bundle L over X together with a lift of the π-action to the total space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Given such a lift, there exists a set of isotropy representations of the π-action on each fiber over the fixed point set which we denote by L | xi = tλi over isolated fixed points and L | Fj = tλj over a fixed 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Denote the collection of these isotropy representations by I = {λi, λj | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' With this notation we have our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) denote a simply-connected, closed 4-manifold with a locally linear, homologically trivial action of a cyclic group π = Z/p of odd prime order p, with fixed-point rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A collection I of integers {λi, λj | i ∈ I, j ∈ J} can be realized (modulo p) as the isotropy representations of an equivariant line bundle (L, π) → (X, π) if and only if there is a collection of integers {mj | j ∈ J} such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) � i∈I λi aibi + � j∈J cjmj − λjαj c2 j ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' When a solution exists, the integers mj = c1(i∗L)[Fj] satisfy equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A special case of this result can be found in [22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4], for a cyclic group π of odd order acting locally linearly and semi-freely on the complex projective plane CP 2 with has three isolated fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The necessary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) in Theorem A is established by extending the methods of [22, §2] to more general actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The definitions above generalize directly to actions (X, π) where π is any finite group, and the structural group G of the principal bundle is any compact Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For applications in gauge theory, G = SU(2) is an important example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The details will be left to the reader (see also [18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Some motivating questions Here are some questions related to the general theme (all actions will be assumed to preserve orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' More information about some of these directions can be found in the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Does there exist a smooth Z/p-action on a (homotopy) K3 surface, which induces the identity on integral homology ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This is a well-known question of Allan Edmonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that results of Edmonds and Ewing [12] imply that topological locally linear examples exist for odd primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Does there exist a smooth Z/p-action on a homotopy K3 surface, which contains an invariant embedded Brieskorn homology 3-sphere Σ(2, 3, 7) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 3 Fintushel and Stern [14] showed that many homotopy K3 surfaces admit embeddings of Σ(2, 3, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The question concerns the possible existence of an equivariant splitting of the K3 surface along the Brieskorn sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A Brieskorn homology 3-sphere Σ(a, b, c) admits a free Z/p-action if p ∤ abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Does there exist a smooth, homologically trivial extension of this action with isolated fixed points to any smooth simply connected negative definite 4-manifold X with boundary Σ(a, b, c) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Anvari [1] proved that the free Z/7 on Σ(2, 3, 5) does not extend in this way over the min- imal negative definite 4-manifold obtained by resolving the link singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The method of proof involves studying equivariant Yang-Mills gauge theory on the non-compact manifold with cylindrical end obtained from X by attaching the end Σ(2, 3, 5)×[0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Information about the Floer homology of Brieskorn spheres is an essential ingredient in tackling this problem (see Saveliev [32, 33, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' What sets of rotation numbers can be realized by a smooth, pseudo-free Z/n-action on X = S2 × S2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' A pseudo-free action is one with isolated singular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If the action is semi-free, there are “standard models” with rotation data {(a, b), (c, d), (a, −b), (c, −d)} at the four fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This question for X = CP 2 was answered in [11, 20], where it turned out that the rotation data was the same for locally linear actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) denote a smooth action of a finite group π on a closed, simply connected smooth 4-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Under what conditins does there exist a π-equivariant principal G- bundle over X with prescribed Chern classes, for G = U(1) or G = SU(2) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Some information about this question was provided in [22, 20] for X = CP 2 and π finite cyclic, or more generally for X negative definite (see also [19] for the connection between Chern classes and the isotropy representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For X = S4, the existence and classification of such bundles was applied by Austin [4] and Furuta [15, 17, 16] to study group actions via instanton gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The compactification of an equivariant version of Donaldson’s Yang-Mills moduli space [8], [20] involves “bubbling” convergence to equivariant instantons over the 4-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Further applications of equivariant bundles arise in studying the equivariant compactification of moduli spaces over cylindrical end 4-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The G-Signature Formula In this section we review the terms of the G-signature formula that we will need in deriving congruence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The G-signature of a closed 4-manifold X with an orientation preserving action of a finite group G acting as isometries on X is defined as the virtual representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) Sign(X, G) = [H2 +(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C)] − [H2 −(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C)] where H2 ±(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C) are the maximal positive/negative definite G-invariant subspaces of H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Taking characters gives the g-signatures (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2) Sign(g, X) = trg H2 +(X) − trg H2 −(X) 4 NIMA ANVARI AND IAN HAMBLETON which by the G-signature formula can be computed from the fixed point set Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let D : C∞(Λ+) → C∞(Λ−) denote the signature operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The Lefschetz numbers are computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3) L(g, D) = (−1)n(n+1)/2 chg(Λ+ − Λ−)(TX | Xg ⊗ C)Td(TXg ⊗ C) e(TXg) chg(Λ−1Ng ⊗ C) [Xg] Where n = dim Xg and note that TX | Xg = TXg ⊕ Ng and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4) chg(Λ+ − Λ−)(TX | Xg ⊗ C) = chg(Λ+ − Λ−)(TXg ⊗ C) chg(Λ+ − Λ−)(Ng ⊗ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let F denote a fixed surface, then the contribution to the g-signature is given by L(g, D) | F = (−1)(e−x − ex)(e−y−iθ − ey+iθ) x(1 − ey+iθ)(1 − e−y−iθ) x(−x) (1 − e−x)(1 − ex)[F] = coth �y + iθ 2 � x cot �x 2 � [F] where the following trigonometric identity is used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5) coth �x 2 � = e−x − ex (1 − e−x)(1 − ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To evaluate on [F], we use the Taylor expansions x coth (x/2) = 2 + 1/6x2 + · · · coth �y + iθ 2 � = coth (iθ/2) − 1 2 csch2 �iθ 2 � y Thus the contribution to the Lefschetz number is given by L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}[F] = − csch2(iθ/2)[F]2 = csc2(θ/2)[F]2 = −4tcF (tcF − 1)2[F]2 where θ = 2πcF p and cF is the rotation number on the normal fiber of F and t = e2πi/p is a primitive pth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly we can compute the contribution from isolated fixed points: L(g, D) | pt = (e−iθ1 − eiθ1)(e−iθ2 − eiθ2) (1 − eiθ1)(1 − e−iθ1)(1 − eiθ2)(1 − e−iθ2) = coth(iθ1/2) coth(iθ2/2) = − cot(θ1/2) cot(θ2/2) = (ta + 1)(tb + 1) (ta − 1)(tb − 1) FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 5 where θ1 = 2πa p and θ2 = 2πb p are the rotation numbers at the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By summing over the fixed point set the G-signature formula is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6) Sign(X) = � i (tai + 1) (tai − 1) (tbi + 1) (tbi − 1) + � j −4αjtcj (tcj − 1)2 where αj denotes the self-intersection [Fj] · [Fj] of the fixed 2-spheres {Fj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This formula can be viewed as an equation in the cyclotomic field Q[ζ] = Q[t]/Φp(t) where Φp(t) is the cyclotomic polynomial 1 + t + t2 + · · · + tp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Congruence Relations for the G-Signature Formula In this section we derive congruence relations satisfied by the rotation data F using the G-signature formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first note that since (ta−1)/(t−1) is a unit in ring of cyclotomic integers Z[ζ] when (a, p) = 1, multiplying both sides of the G-signature formula by (t−1)2 induces an equation in the ring R = Z[ζ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The I-adic expansion of the resulting right- hand side leads to congruence relations relating the rotation data, where I denote the ideal generated by (t − 1) in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Following the method of [22] we lift the equation to Z[t], compute the Taylor expan- sions about t = 1 and reduce the coefficients modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since the indeterminacy of the coefficients are determined from the expansion of the cyclotomic polynomial Φp(t) (for which p divides the coefficients of its Taylor expansion about t = 1 up to order p − 1) we obtain valid congruence relations by equating coefficients modulo p up to order p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The expansion arising from contributions from isolated fixed points are given by (ta + 1) (ta − 1) (tb + 1) (tb − 1)(t − 1)2 = 4 ab + 4 ab(t − 1) + 1 3 �a2 + b2 + 1 ab � (t − 1)2 − 1 180 �a4 + b4 − 5a2b2 + 3 ab � (t − 1)4 + · · · Similarly the expansion of the second term is given by expressions of the form −4αtc (tc − 1)2(t − 1)2 = −4α c2 + −4α c2 (t − 1) + 1 3 α(c2 − 1) c2 (t − 1)2 − 1 60 α(c − 1)(1 + c + c2 + c3) c2 (t − 1)4 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equating both sides of the expansion (mod p) from the resulting equation in R we thus obtain the following congruence relations: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' [22, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 625] Let (X, π) denote a simply connected, closed 4-manifold with a homologically trivial, locally-linear group action of a finite cyclic group π = Z/p of odd 6 NIMA ANVARI AND IAN HAMBLETON order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the following congruence relations hold (1) � i 1 aibi − � j αj c2 j ≡ 0 (mod p) (2) � i a2 i + b2 i aibi + � j αj ≡ 3 Sign(X) (mod p) (3) � i a4 i + b4 i − 5a2 i b2 i aibi + 3 � j αjc2 j ≡ 0 (mod p) (4) � i 2a6 i − 7a4 i b2 i − 7a2 i b4 i + 2b6 i aibi + 10 � j αjc4 j ≡ 0 (mod p) Higher-order relations are valid up to and including terms of order p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2 (Linear models on CP 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let G = Z/p with odd prime p act linearly on X = CP 2 by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] for 0 < a < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The fixed point set consists of one isolated fixed point [1 : 0 : 0] with tangential rotation number (a, a) and a fixed projective line F = {[z1 : z2 : z3] | z1 = 0} with self-intersection +1 and a rotation of cF ≡ a (mod p) on the normal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then it is easy to check that the congruence relations are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly, in the case when the action is given by t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p the action consists of three isolated fixed points [0 : 0 : 1], [1 : 0 : 0], [0 : 1 : 0] with rotation numbers (a, b), (b − a, −a), (a − b, −b) and with some algebra it can be checked that the congruence relations hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Additional examples can be obtained for #nCP 2 by equivariant connected sums along fixed point sets using the linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equivariant Line Bundles Let (X, π) denote a simply connected, closed 4-manifold with a homologically trivial action of a finite cyclic group π = Z/p of odd prime p and L → X an equivariant line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We compute the contribution of a fixed surface F to the twisted G-signature formula: L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}{chg(i∗L)}[F] = {2 coth(iθ/2) − csch2(iθ/2)y}{ez+iφ}[F] = {2 coth(iθ/2) − csch2(iθ/2)y}{eiφ + eiφz}[F] = {2 coth(iθ/2)eiφz − csch2(iθ/2)yeiφ = 2c1(i∗L)[F](tcF + 1) (tcF − 1)tλ + −4[F]2tcF (tcF − 1)2 tλ where φ = 2πλ p and z = c1(i∗L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly for the contribution to the isolated fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To summarize, given an equivariant line bundle (L, π) → (X, π), the index of the twisted G-signature operator gives a virtual character given by FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 7 χ(t) = � i (tai + 1) (tai − 1) (tbi + 1) (tbi − 1)tλi + � j −4αjtcj (tcj − 1)2tλj + � j 2c1(i∗L)[Fj](tcj + 1) (tcj − 1)tλj where {Fj} are fixed 2-spheres of the action on X with αj denoting the self-intersection [Fj] · [Fj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that χ(1) = ch(L)L(X)[X] = � 1 + c1(L) + 1 2c1(L)2 � � 4 + p1 3 � [X] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) = �p1 3 + 2c1(L)2� [X] = Sign(X) + 2c1(L)2[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2) Since χ(t) is a virtual character for G = Z/p we may write χ(t) = �p−1 i=0 aiti for some ai ∈ Z and χ(t)(t − 1)2 = χ(1)(t − 1)2 + higher order terms (mod p) We can then take the Taylor expansion of the right-hand side of the twisted G-signature formula after multiplying by (t−1)2 and equate the first and second order terms to obtain two additional congruence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The first order term vanishes while the second order term is congruent to χ(1) (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Taking Taylor expansions for these terms in the G-signature formula give: (ta + 1) (ta − 1) (tb + 1) (tb − 1)(t − 1)2tλ = 4 ab + 4(λ + 1) ab (t − 1)+ 1 3 (a2 + b2 + 1 + 6λ2 + 6λ) ab (t − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly for the second term: −4αtc (tc − 1)2(t − 1)2tλ = −4α c2 + −4α(1 + λ) c2 (t − 1)+ 1 3 α(c2 − 1 − 6λ − 6λ2) c2 (t − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' and for the third term: 2m(tc + 1) (tc − 1)(t − 1)2tλ = 4m c (t − 1) + 2m(2λ + 1) c (t − 1)2 + · · · where m = c1(i∗L)[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Combining these expressions we obtain the following theorem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (L, π) → (X, π) denote an equivariant line bundle over a simply connected, closed 4-manifold with a homologically trivial group action of a finite cyclic 8 NIMA ANVARI AND IAN HAMBLETON group π = Z/p of odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the following congruence relation holds (i) � i λi aibi − � j λjαj c2 j + � j c1(i∗L)[Fj] cj ≡ 0 (mod p) (ii) � i λ2 i aibi − � j λ2 jαj c2 j + 2 � j λjc1(i∗L)[Fj] cj ≡ c1(L)2[X] (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let π = Z/p act on X = CP 2 preserving an almost complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the complexified second exterior power of the tangent bundle is an equivariant line bundle see [22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3 (Linear models on CP 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let p denote an odd prime and consider X = CP 2 and a linear action t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We give an explicit construction of equivariant line bundles over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Consider a finite dimensional complex representation space V = C(λ1) ⊕ C(λ2) ⊕ C(λ3) with action given by ρ ∈ GL3(C): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4) ρ = \uf8eb \uf8ed tλ1 tλ2 tλ3 \uf8f6 \uf8f8 : C3 \\ {0} −→ C3 \\ {0} with the λi positive integer weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let S(V ) denote the unit sphere in V then ρ com- mutes with the free S1-action on S(V ) and CP 2 = S(V )/S1 = S5/S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The ρ-action on S(V ) is a lift of the linear action on X if the following system of linear congruences λ1 − λ3 ≡ a, λ2 − λ3 ≡ b λ2 − λ1 ≡ b − a, λ3 − λ1 ≡ −a λ1 − λ2 ≡ a − b, λ3 − λ2 ≡ −b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This system has one degree of freedom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' let λ3 = λ be a fixed parameter, then the isotropy representations over the three isolated fixed points are given by: fixed point p1 = [0 : 0 : 1], rotation number (a, b), with isotropyλ3 ≡ λ fixed point p2 = [1 : 0 : 0], rotation number (b − a, −a), with isotropyλ1 ≡ λ + a fixed point p3 = [0 : 1 : 0], rotation number (−b, a − b)with isotropyλ2 ≡ λ + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The equivariant line bundle L = S(V ) ×S1 C is the canonical bundle over CP 2 and the congruence relations of the theorem are satisfied: � i λi aibi ≡ 0 � i λ2 i aibi ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 9 In the case when b = 0 the action is given by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] and has a fixed projective line F = {z1 = 0} with normal rotation number cF ≡ a (mod p) and self-intersection +1, while the isolated fixed point [1 : 0 : 0] has rotation number (a, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The compatibility for a lift of the linear action is the congruence relation: λ − λF ≡ a (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It is easily seen that the congruence relation of the theorem are satisfied: λ a2 − λF a2 + c1(i∗L)[F] a ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' where we used c1(i∗L)[F] = −1 since the first Chern class of the canonical line bundle over CP 2 is negative of the preferred generator [F] ∈ H2(CP 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) [30, Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Similarly, the second relation is λ2 a2 − λ2 F a2 + 2λFc1(i∗L)[F] a ≡ (a + λF)2 − λ2 F a2 + 2λFc1(i∗L)[F] a ≡ 1 ≡ c1(L)2[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (X, π) be a homologically trivial of π = Z/p in the setting of Theorem A, with rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We say that (X, π) satisfies the condition of Theorem A if there exists a set of isotropy data I = {λi, λj | i ∈ I, j ∈ J}, and a set of integers {mj | j ∈ J} so that the equation given in Theorem A holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the next statement, we apply the equivariant connected sum operation to line bun- dles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Suppose that (X, π) satisfies the condition of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' There there is an equivariant connected sum (X♯ CP 2, π) with a linear π-action on CP 2 which also satisfies the condition of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first suppose that (X, π) contains a fixed 2-sphere Fj with data {αj, cj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (CP 2, π) be the linear action given by t · [z1 : z2 : z3] = [ζ−cjz1 : z2 : z3] as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We do the equivariant connected sum (preserving the orientations) along a point in Fj and a point in the fixed 2-sphere of CP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The new data is obtained by (i) adding the data {(−cj, −cj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' λj} for the newly created isolated fixed point (on CP 2), and (ii) the data {(cj, αj + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' λj} for the new fixed 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' With these choices, it follows that the action (X♯ CP 2, π) satisfies the condition of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The proof in case the action (X, π) has only isolated fixed points is easier, and will be left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Suppose that (X, π) has data satisfying the condition of Theorem A, and contains a fixed 2-sphere F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X0 ⊂ X denote the complement of a linear π-invariant 4-ball neighbourhood of a point x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If L is an equivariant line bundle over (X♯ CP 2, π), then the restriction of L to X0 extends to an equivariant line bundle over (X, π) realizing the given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 10 NIMA ANVARI AND IAN HAMBLETON 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The proof of Theorem A The first relation in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1 proves the necessary conditions of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To prove sufficiency we will need the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that in a standard lens space Y = L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b), a generator µ ∈ H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is repre- sented by a circle fibre in the fibration S1 → L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) → S2 given by the quotient of a free Z/n action on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The linking paring lk: H1(Y ) × H1(Y ) → Q/Z in the lens space Y = L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) is given by lk(µ, µ) = ab p where µ is a generator of H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = Z/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the usual representation of lens spaces L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' q) as the quotient of the free Z/n action t · (z1, z2) = (ζz1, ζqz2) on S3, the linking pairing is given by lk(µ, µ) = q n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The diffeomorphism L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a∗b) → L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) arising from changing the generator in Z/n induces a map on first homology given by multiplication by a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It follows that the linking pairing on Y is given by lk(aµ, aµ) = a2 · �a∗b n � = ab n ∈ Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2 ((See [22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 621], [25, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 95])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let φ: π1(Y ) −→ U(1) be the holonomy representation of a flat U(1)-bundle over the lens space Y = L(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' a, b) that sends a generator µ to exp(2πiλ/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the Poincar´e dual of the first Chern class PD(c1(L)) is given by λ ab[µ] in H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The adjoint to the linking form Φ: H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) → Hom(H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z), Q/Z) sends m[µ] to lk(mµ, −) which can be identified with the holonomy representation of the flat bundle via : e2πi·lk(mµ,−) : H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) → U(1) and this maps the generator to exp(2πi · mab/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It follows that m ≡ λ ab (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ If Y is a lens space, we let ˆµ ∈ H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) denote a standard cohomology generator: the Poincar´e dual to the circle fibre class µ ∈ H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let u: Y ′ → Y be a d-fold regular covering of lens space, where H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/dn and H1(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/n, with gcd(d, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then u∗(ˆµ) = ˆµ′ ∈ H2(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For a d-fold regular covering u: Y ′ → Y of lens spaces, we have u∗[µ′] = d[µ] ∈ H1(Y, Z) and u∗[Y ′] = d[Y ] ∈ H3(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The cohomology generator ˆµ = [Y ] ∩ µ ∈ H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is the Poincar´e dual of µ, and similarly for ˆµ′ ∈ H2(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We have the formula u∗([Y ′] ∩ u∗(ˆµ)) = u∗(µ′) = dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = Z/dn and H1(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = Z/n, where gcd(d, n) = 1, then u∗(ˆµ) = kˆµ′ implies that k ≡ 1 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Hence u∗(ˆµ) = ˆµ′ ∈ H2(Y ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 11 Suppose that (X, π) satisfies the assumptions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let Σ ⊂ X denote the singular set of the action, and let X0 := X − Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Write αj = pajαj, where αj is prime to p (for each fixed 2-sphere Fj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If the singular set Σ ⊂ X contains an isolated point, then H1(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is a quotient of � Z/αj and has order prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let F ⊂ X denote the disjoint union of the fixed 2-spheres, so F = � j Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' First note that H1(X0) ∼= H3(X, F) and we have an exact sequence · · → H2(X) → H2(F) → H3(X, F) → H3(X) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since H3(X) = 0, and the homology classes of the fixed 2-spheres are linearly independent mod p (by [9, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6]), it follows that H3(X, F) ∼= H1(X0) is a torsion group of order prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Moreover, the exact Mayer-Vietoris sequence 0 → H2(X0) ⊕ H2(F) → H2(X) → H1(∂X0) → H1(X0) → 0 and the equality H1(∂X0) = H1(∂ν(F)) ∼= � Z/αj completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ The proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1(i), the indicated formulas hold if (X, π) admits an equivariant line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' It remains to prove that a solution {λi, λj, mj | i ∈ I, j ∈ J} to the congruence relation � i λi aibi − � j λjαj c2 j + � j mj cj ≡ 0 (mod p) is sufficient for the existence of an equivariant line bundle with {λi, λj} isotropy repre- sentations over the isolated fixed points and 2-spheres respectively, and mj ≡ c1(i∗L | Fj) mod αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For simplicity, we will assume that the action (X, π) contains at least one iso- lated fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This may always be arranged by taking the equivariant connected sum of (X, π) along a fixed 2-sphere with a suitable linear π-action on CP 2 (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X0 = X−N, where N = ν(Σ) is a π-invariant tubular neighbourhood of the singular set Σ ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' More explicitly, X0 is the compact 4-manifold with boundary obtained by removing π-invariant 4-balls around each isolated fixed point and π-invariant tubular neighbourhoods D2 → ν(Fj) → Fj around each π-fixed 2-sphere Fj with rotation tcj on D2-fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then ν(Fj) is a 2-disk bundle over S2 with Euler class αj[F] ∈ H2(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z, and the lens space ∂ν(Fj) = L(αj, 1) inherits a free Z/p action with rotation number cj on the circle fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If W0 := X0/π denotes the quotient manifold with (regular) covering map q: X0 → W0 classified by u: W0 → Bπ, then the boundary ∂W0 consists of lens spaces Yi = L(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' ai, bi) and Yj = L(pαj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' cj, cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that H1(W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/p ⊕ H1(X0, Z) by the spectral sequence of the covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4, H1(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is a quotient of � Z/αj and has order prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Recall that π-equivariant line bundles L over (X, π) are classified by an element θ(L) ∈ H2 π(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = H2(X ×π Eπ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) 12 NIMA ANVARI AND IAN HAMBLETON in the Borel requivariant cohomology of X (see [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since the π-action on X0 is free, for the restriction L0 ց X0 we have θ(L0) ∈ H2 π(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= H2(W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z), and θ(L0) = c1(¯L0), where ¯L0 is the line bundle over W0 obtained by dividing out the free π-action on the total space of L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Moreover, in the short exact sequence 0 → H2(Z/p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) c∗ −→ H2(W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) q∗ −→ H2(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) → 0 the pullback q∗(θ(L0)) = c1(q∗(¯L0)) = c1(L0) ∈ H2(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The strategy will be to find a suitable element θ(L) ∈ H2 π(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) by studying the Mayer- Vietoris sequence · · → H2 π(X) → H2 π(X0) ⊕ H2 π(N) → H2 π(∂X0) δ−→ H3 π(X) → H3 π(X0) ⊕ H3 π(N) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' in Borel cohomology associated to the π-equivariant decomposition X = X0 ∪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We observe the following: (i) The Mayer-Vietoris coboundary map H2 π(∂X0) δ−→ H3 π(X) factors H2 π(∂X0) δ−→ H3 π(X0, ∂X0) ∼= H3 π(X, N) → H3 π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (ii) The cokernel of the map H2 π(N) → H2 π(∂X0) has exponent p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This follows from the commutative diagram of restriction maps H2 π(N) � � H2 π(∂X0) � � � Z/pαj � H2(N) � H2(∂X0) ∼ = � � Z/αj since the map H2 π(N) → H2(N) is surjective (by the Borel spectral sequence) and the map H2(N, ∂N) → H2(N) is adjoint to the (diagonal) intersection form on N, with cokernel H2(∂N) = H2(∂X0), hence determined by the self-intersection numbers {αj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' (iii) We have H3 π(X0, ∂X0) ∼= H3(W0, ∂W0) ∼= H1(W0) ∼= Z/p⊕H1(X0), where H1(X0) is a quotient of � Z/αj and has order prime to p (by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' To complete the proof of Theorem A, it is now enough to produce a class θ0 ∈ H2 π(X0) ∼= H2(W0), which added together with the classes already found in H2 π(N) will have image zero in H2 π(∂X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By the observations above, this amounts to finding a U(1)-bundle ¯L0 on ∂W0 whose first Chern class θ0 = c1(¯L0) has image of order prime to p under the coboundary map H2(∂W0) → H3(W0, ∂W0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In other words, we need to find suitable U(1)-bundles over each of boundary components of ∂W0, so that the sum of their first Chern classes is zero (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This required relation is exactly the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) given in the statement of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let Y denote one of the lens spaces in ∂W0, For convenience, we will identify Z/p ∼= H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) by a �→ a · ˆµ, for a ∈ Z/p, where ˆµ ∈ H2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) denotes a standard generator, Poincar´e dual to the circle fibre class µ ∈ H1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) (introduced in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 13 If Y = L(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' ai, bi) then choose the holonomy representation that sends a generator in π1(Y ) to exp(2πiλi/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The first Chern class of the associated flat U(1)-bundle is λi aibi [ˆµ] ∈ H2(Yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the case when the π-action on X has only isolated fixed points, the condition for an extension is that these elements lie in the kernel of δ in H2(∂W0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) = � i H2(Yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) which is equivalent to the condition � i λi aibi ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the general case when components of the fixed set contain 2-spheres, we need to consider contributions from the lens spaces Yj = L(pαj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' cj, cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' These lens spaces arise from the free tcj-action on �Yj := ∂ν(Fj) ≈ L(αj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Conisder the induced covering spaces of Yj by the lens spaces L(paj+1, 1) and L(αj, 1), where αj = pajαj, and note that the covering maps induce an isomorphism: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5) H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼ = � ∼ = � H2(L(paj+1, 1)) ⊕ H2(L(αj, 1)) ∼ = � Z/pαj ∼ = � Z/paj+1 ⊕ Z/αj Under the two covering maps, the standard cohomology generator ˆµj ∈ H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) is sent to the standard generators in H2(L(paj+1, 1)) and H2(L(αj, 1)), respectively, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The maps in the lower sequence are the reductions mod paj+1 and α, after using the identifications provided by the cohomology generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Now the following congruences uniquely determines a Chern class c1(¯Lj) = ℓj c2 j [ˆµj] ∈ H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z), by choosing: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='6) ℓj ≡ −λjαj (mod paj+1), ℓj ≡ cjmj (mod αj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' and hence a U(1)-bundle ¯Lj ց Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The minus sign is chosen in the first congruence because the induced orientation on Yj from ∂W0 is opposite to its orientation as the disk bundle over S2 with Euler class αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' By diagram (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3, the first Chern class has image c1(¯Lj) = ℓj c2 j [µj] = −λjαj + cjmj c2 j [µj] ∈ H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) with respect to the decomposition H2(Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Z) ∼= Z/paj+1 ⊕ Z/αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' After substituting these expressions into the formula of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1, we see that the sum vanishes mod p, and hence the required line bundle ¯L0 over W0 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equivariant SU(2) Bundles In this section we compute a (necessary) congruence relation similar to the previous section, but for equivariant SU(2)-bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' As above, we work over a closed, simply connected, oriented 4-manifold with a finite homologically trivial cyclic group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We 14 NIMA ANVARI AND IAN HAMBLETON again use the twisted G-signature formula (with the previously established notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In particular, let D denote the signature operator twisted by an equivaraint SU(2)-bundle E −→ X, then the contribution to the Lefschetz numbers from isolated fixed points is given by L(g, D) | pt = (ta + 1) (ta − 1) (tb + 1) (tb − 1)(tλ + t−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We need to compute the contribution from isolated fixed 2-spheres F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since E | F = L ⊕ L−1, we have chg(L ⊕ L−1 | F) = {eλ+z + e−λ−z}[F] and L(g, D) | F = {2 cot(iθ/2) − csch2(iθ/2)y} chg(L ⊕ L−1 | F)[F] = {2(tc + 1) (tc − 1) − 4tcy (tc − 1)2}{eλ(1 + z) + e−λ(1 − z)}[F] = {2(tc + 1) (tc − 1) − 4tcy (tc − 1)2}{tλ + t−λ + z(tλ − t−λ)}[F] = − 4tc[F]2 (tc − 1)2(tλ + t−λ) + 2c1(L)[F](tc + 1) (tc − 1)(tλ − t−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Also note χ(1) = ch(E)L(X)[X] = (2 − c2(E))(4 + 1 3p1) = 2 Sign(X) − 4c2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We now again multiply both sides of the G-signature formula by (t − 1), take Taylor expansions about t = 1 and reduce coefficients modulo p: (ta + 1) (ta − 1) (tb + 1) (tb − 1)(t − 1)2(tλ + t−λ) = 8 ab + 8 ab(t − 1)+ 2 3 (a2 + b2 + 1 + 6λ2) ab (t − 1)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' and for the second term, where we let m denote c1(L)[F]: (t − 1)2{ −4αtc (tc − 1)2(tλ + t−λ) + 2m(tc + 1) (tc − 1)(tλ + t−λ)} = −8α c2 + −8α c2 (t − 1) + 2 3 (αc2 − α − 6αλ2 + 12mcλ) c2 (t − 1)2 + · · · Summing over all the fixed sets and simplifying the coefficient of second order term (t−1)2, we obtain: 2 Sign(X) + � i 4λ2 i aibi − � j 4αjλ2 j c2 j + � j 8mjλj cj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equating this with χ(1) = 2 Sign(X) − 4c2(E) and reducing coefficients modulo p gives the following congruence relation: FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 15 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let (E, π) → (X, π) denote an equivariant SU(2)-bundle over a simply connected, closed 4-manifold with a homologically trivial group action of a finite cyclic group π = Z/p of odd prime order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the following congruence relation holds � i λ2 i aibi − � j αjλ2 j c2 j + � j 2λj cj c1(i∗Lj)[Fj] ≡ −c2(E)[X] (mod p), where Lj is a local reduction E | Fj = Lj ⊕ L−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='2 (Linear Models on S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X = S4 with a linear Z/p-action which gives rotation numbers (a, b) and (a, −b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let E denote the instanton one equivariant SU(2)- bundle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' with c2(E) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Then the congruence relation is given by −c2(E) = λ2 1 ab − λ2 2 ab (mod p) It is elementary to check that for congruence relation is satisfied with the following isotropy representations λ1 = b − a 2 λ2 = a + b 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' over the fibres of the fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3 (Linear Models on CP 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X = CP 2 with a linear Z/p-action with one isolated fixed point with rotation number (a, −a) for some a (mod p) and a fixed projective line F with rotation number a (mod p) on the normal bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let E again denote the instanton one equivariant SU(2)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The congruence relation gives −1 ≡ −λ2 a2 + λ2 F a2 + 2mλF a where m = c1(i∗L)[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' There exists two distinct lifts giving rise to equivariant bundles which admit G-invariant ASD connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the case when the equivariant lift comes from ”bubbling” on the isolated fixed point then m = 0 and λ ≡ a (mod p) λF ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Thus the congruence is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' On the other hand, if we choose the equivariant lift associated to the fixed 2-sphere (from 3-dimensional fixed connected component in the moduli space of equivariant ASD connections with c2(E) = 1) then m = −1 and λ ≡ a/2 (mod p) λF ≡ a/2 (mod p), again the congruence relation is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the next section we compute the dimension of the moduli space of invariant anti-self dual connections for a given equivariant SU(2)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 16 NIMA ANVARI AND IAN HAMBLETON 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Equivariant Index Computation Let X be a simply connected, closed, smooth negative definite 4-manifold, with a homologically trivial action of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' If E ց X is an SU(2)-bundle with c2(E) = k, the moduli space M∗ 1(X) of irreducible ASD connections (on an SU(2)-bundle E with c2(E) = 1) inherits a G-action, and the connected components of the fixed point set MG 1 (X) correspond to G-invariant ASD connections for certain equivariant lifts of the G-action on X to E (see [15], [6], [20, Theorem A], [24, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We want to compute the dimension of the moduli space MG k (X) of irreducible G- invariant ASD connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This is motivated by the following example, for which the formal dimension dim M∗ 1(X) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In this case, we expect a dimension formula that gives 1 and 3-dimensional strata depending on contributions from isolated fixed points or isolated fixed 2-spheres in X and on the isotropy representations from the equivariant lift (see [6] and [21] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' There are similar index calculations in the literature in various gauge-theoretic settings (for example, see [13, §3], [4]), [28, 29], [35], [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first very briefly review the dimension calculation in the non-equivariant setting to set some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let D+ A = d∗ A + d+ A : Ω1(ad E) → Ω0(ad E) ⊕ Ω2 +(ad E) denote the anti-self duality operator, and let Mk denote the ASD moduli space with c2(E) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Note that the formal dimension is given by dim Mk = − Ind(D+ A) and this is given by (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='1) Ind(D+ A) = ˆA(X) ch(S+) ch(adC E)[X] where S = S+ ⊕S− and ˆA(X) = � xi/2 sinh(xi/2) with ch(S) = �(exi/2 + e−xi/2) and ch(S+) − ch(S−) = �(exi/2 − e−xi/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Using this we compute 2 ˆA(X) ch(S+) ch(adC E)[X] = (4 + 1 3p1 + χ)(3 − 4c2(E))[X] = −16c2(E) + 3(p1 3 + χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Thus the index Ind(D+ A) = −8c2(E) + 3 2(Sign +χ)(X) and we get the usual expression dim Mk = 8k − 3/2(χ + Sign)(X) for the dimension of the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Also note the following alternative expression for the index: Ind(D+ A) = ch(S+ − S−) ch(S+) ch(adC E)Td(TX ⊗ C) e(X) [X] = ˆA(X) ch(S+ ⊗ adC E)[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For the equivariant setting E is an equivariant SU(2)-bundle and let D = D+ A denote the anti-self duality operator d∗ A + d+ A : Ω1(ad E)G → Ω0(ad E)G ⊕ Ω2 +(ad E)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We compute the equivariant index by averaging the Lefschetz numbers as in [13]: Ind(D) = 1 p � g∈G L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D) FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 17 Ind(D) = 1 p{L(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D) + � g̸=1 L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D)} = 1 p{−8c2(E) + 3 2(χ + Sign)(X) + � g̸=1 L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D)} = 1 p{−8c2(E) + 3p 2 (χ + Sign)(X/G) − 3 2(dχ + dσ)(XG) + � g̸=1 L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' D)} where pχ(X/G) = χ(X) + dχ with dχ = � g̸=1 χ(Xg) is the Euler characteristic defect terms and similarly for the signature defect term: − 3 2(dχ + dσ)[pt] = −3 2(1 − cot(θ1/2) cot(θ2/2)) − 3 2(dχ + dσ)[F] = −3 2(2 + [F]2 csc2(θ/2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' where (θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' θ2) are the rotation numbers at an isolated fixed point and θ = cF is the rotation number on the normal bundle to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Decomposing the contributions from isolated fixed points and 2-spheres: � g̸=1 L(g, D)(XG) = � g̸=1 { � i L(g, D) | (ai,bi) + � j L(g, D) | Fj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Now chg(adC E)(pt) = 3 − 4 sin2( πkℓ p ), with ℓ the isotropy representation on the fiber of E over the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The Lefshetz numbers from the fixed sets can be computed directly from the index formula and are given by: (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='3) L(g, D) | pt = −1 2 [cot(θ1/2) cot(θ2/2) − 1] chg(adC E)[pt] (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='4) L(g, D) | F = [−i cot(θ/2) + 1 2(χ + csc2(θ/2)y)] chg(adC E)[F], with χ the Euler class of the tangent bundle to F and y is the Euler class of the normal bundle to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We first compute the contribution from isolated fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' L(g, D) | pt = −1 2 [cot(θ1/2) cot(θ2/2) − 1][3 − 4 sin2(πkℓ p )][pt] = −3 2[cot(θ1/2) cot(θ2/2) − 1] − 2 sin2(πkℓ p ) + 2 cot(θ1/2) cot(θ2/2) sin2(πkℓ p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 18 NIMA ANVARI AND IAN HAMBLETON Summing over all isolated fixed points gives 1 p � g̸=1 � i L(g, D) | (ai,bi) = 3 2p � i (dχ + dσ)(ai, bi) − 2 p � i p−1 � k=1 sin(πkℓi p ) + 2 p � i p−1 � k=1 cot(aiπk p ) cot(biπk p ) sin2(πkℓi p ) = 3 2p � i (dχ + dσ)(ai, bi) + m + � i ρL(p, ai, bi, ℓi) where m is the number isolated fixed points with non-trivial representation on the fiber and ρL(p, a, b, ℓ) is the rho invariant of lens spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We need to compute chg(adC E | F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since an SU(2) bundle restricted over a fixed 2- submanifold has a local abelian reduction E | F = L ⊕ L−1 for some L, we have ad E | F = L2 ⊕ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We need to compute chg(adC E | F) = chg(L2) + chg(L2) + 1 and this contributes chg(adC E | F) = (g + gc1(L2)) + (g−1 + g−1c1(L2)) + 1 = (g + g−1 + 1) + c1(L2)(g − g−1) = (3 − 4 sin2(πkℓ p )) + 2ic1(L2) sin(2πkℓ p ), where now ℓ is the isotropy representation on the fibre over the fixed 2-sphere F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Substi- tuting these terms, the Lesfchetz number L(g, D) | F evaluated on fixed 2-spheres gives: L(g, D) | F = [−i cot(θ/2) + 1 2(χ + csc2(θ/2)y)][(3 − 4 sin2(πkℓ p )) + 2ic1(L2) sin(2πkℓ p )][F] = 1 2[χ + csc2(θ/2)y][3 − 4 sin2(πkℓ p )] + 2c1(L2) sin(2πkℓ p ) cot(θ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let us introduce a kind of rho invariant term for fixed surfaces: ρF(ℓ) = 2 p p−1 � k=1 csc2(πcFk p ) sin2(πkℓ p )[F]2 − 4c1(L)[F] p p−1 � k=1 sin(2πkℓ p ) cot(πkcF p ), with this notation we have 1 p � g̸=1 � j L(g, D) | Fj = 3 2p � j (dχ + dσ)[Fj] − 2 p � j χ(Fj) p−1 � k=1 sin2(πkℓj p ) − � j ρFj(ℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES 19 Now combining all the terms we obtain: Ind(DA) = −8 p c2(E) + 3 2(χ + Sign)(X/G)) − m + � i ρL(p, ai, bi, ℓi) − � j with ℓj̸=0 χ(Fj) − � j ρFj(ℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Since dim MG k (X) = − Ind(DA), the dimension formula is dim MG k (X) = 8 pc2(E) − 3 2(χ + Sign)(X/G) + m − � i ρL(p, ai, bi, ℓi) + � j with ℓj̸=0 χ(Fj) + � j ρFj(ℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Before giving an example we note a few special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' When the action on X only has isolated fixed points, let (ai, bi) denote the rotation numbers and ℓi the isotropy representation over the points, the formula reduces to the following: dim MG k (X) = 8c2(E) p − 3 2(χ + Sign)(X/G) + m − � i ρL(p, ai, bi, ℓi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For invariant ASD connections on the four-sphere this formula reduces to that of [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 394].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' In the case of SO(3)-bundles in the orbifold setting, see Fintushel and Stern [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' When the action on X is a smooth involution with fixed 2-sphere and non-trivial action on fibre cF ≡ ℓ ≡ 1 mod 2 the formula above reduces to: dim MG k (X) = 4c2(E) − 3 2(χ + Sign)(X/G) + χ(F) + [F]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' which matches with Wang [35, Theorem 18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' 130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We finish this section with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Let X = #3CP 2 with a linear Z/p-action with p = 5 that arises from equivariant connected sums of linear actions in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Take the equivariant connected sum of two copies of CP 2 along the two dimensional fixed sets which fixes a pro- jective line and a rotation number of (1, −1) at the isolated fixed points in each copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Now at one of the isolated fixed points take the equivariant connected sum with CP 2 that has a linear action with 3 isolated fixed points with rotation numbers (1, 1), (2, −1), (2, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The result is a smooth, homologically trivial Z/5-action on X that has 3 isolated fixed points with rotation data {(1, −1), (2, −1), (2, −1)} and a single fixed 2-sphere F with rotation number cF ≡ 1 (mod p) on the normal bundle and has self intersection −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The compactified, equivariant ASD instanton one moduli space M1(X) has dimension 5 with fixed components that are 1 and 3-dimensional which correspond to invariant ASD connections for a lifted action to the SU(2)-bundle (see [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The boundary of the moduli space is the ”bubbling” of highly concentrated ASD con- nections which can be identified with a copy of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The isolated fixed points propagate 20 NIMA ANVARI AND IAN HAMBLETON 1-fixed dimensional strata into the moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We will compute the dimension of these strata using the dimension formula from this section and from the fixed point data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' For example, at the isolated fixed point (2, −1) the highly concentrated instantons correspond to ASD connections on the 4-sphere, with equivariant lifts matching the linear models which then pull back to X using the degree 1-map in the formation of the Taubes boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This determines the equivariant lift on X and has isotropy representation tλ1 over the fixed point (2, −1) with λ1 ≡ −3 (mod p) and tλ2 over all the other fixed point sets with λ2 ≡ 1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The dimension formula gives: 8 p − ρL(p, 2, −1, −3) − ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' On the other hand, at a point on the fixed 2-sphere F following the same procedure with the degree one Taubes map, we can pull-back an equivariant bundle from the linear model on S4 with a fixed embedded 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' This time we get an equivariant SU(2)-bundle on X with c1(L)[F] = −1 in the local reduction E | F = L⊕L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' The isotropy representation is tλ over all the fixed point sets with λ ≡ 1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' We then have: 8 p − 2ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' after substituting the data into the dimension formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' References [1] N.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content=' Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton, Ontario, Canada Email address: anvarin@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='ca Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton, Ontario, Canada Email address: hambleton@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'} +page_content='ca' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQfVwQu/content/2301.03105v1.pdf'}