Graphs and $({\mathbb{Z}}_2{)}^k$-actions

Abstract

Let ${\mathcal{A}}_n^k$ denote all nonbounding effective smooth $({\mathbb{Z}}_2{)}^k$-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 ${\mathcal{A}}_n^k$ a $({\mathbb{Z}}_2{)}^k$-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 ${\mathcal{A}}_n^k$, which can become the best possible in some cases; we determine the existence and the equivariant cobordism classification of all actions in ${\mathcal{A}}_n^k(h)$ with $h=3,4$, where ${\mathcal{A}}_n^k(h)$ is the subset of ${\mathcal{A}}_n^k$, each of which is equivariantly cobordant to an effective $({\mathbb{Z}}_2{)}^k$-action fixing just $h$ isolated points, and it is well-known that ${\mathcal{A}}_n^k(h)$ is empty if $h=1,2$; we characterize the explicit relationships among tangent representations at fixed points of each action in ${\mathcal{A}}_n^k(h)$ 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 ${\mathcal{A}}_n^k$.

44 pages with 9 figures. Rewritten and expanded

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