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

Let $\mathcal{A}_n^k$ denote all nonbounding effective smooth $({\Bbb Z}_2)^k$-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 $({\Bbb Z}_2)^k$-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 $({\Bbb Z}_2)^k$-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 $\mathcal{A}_n^k$.

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