Graphs and (\Bbb Z₂)^k-actions

2005/08/31 by Zhi Lü, Lü, Zhi
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.math/0508643

Abstract

Let A_n^k denote all nonbounding effective smooth (\Bbb Z₂)^k-actions on n-dimensional smooth closed connected manifolds, each of which is cobordant to one with finite fixed set. Motivated by GKM theory, one can associate to each action of A_n^k a (\Bbb Z₂)^k-colored regular graph of valence n. Together with the combinatorics of colored graphs, equivariant cobordism and the tom Dieck-Kosniowski-Stong localization theorem, we give a lower bound for the number of fixed points of an action in A_n^k, which can become the best possible in some cases; we determine the existence and the equivariant cobordism classification of all actions in A_n^k(h) with h=3,4, where A_n^k(h) is the subset of A_n^k, each of which is equivariantly cobordant to an effective (\Bbb Z₂)^k-action fixing just h isolated points, and it is well-known that A_n^k(h) is empty if h=1,2; we characterize the explicit relationships among tangent representations at fixed points of each action in A_n^k(h) with h=3,4, which actually give the explicit solution of the Smith problem in such cases. As an application, we also study the minimum number of fixed points of all actions in A_n^k.

Related

• Equivariant geometric bordism, representation, labelled graph
• On the cobordism classification of manifolds with Z/p-action
• (\Bbb Z₂)^k-actions with w(F)=1
• GKM theory for torus actions with non-isolated fixed points
• How is a graph not like a manifold?