On 3-manifolds with locally-standard (Z_2)^3-actions

TL;DR

This paper characterizes which orientable closed 3-manifolds admit locally standard $( ext{Z}_2)^3$-actions, linking topological properties to group actions and extending results to rational homology spheres with specific conditions.

Contribution

It provides a complete classification of certain 3-manifolds with locally standard $( ext{Z}_2)^3$-actions, especially relating to connected sums of $ ext{Z}_2$-homology spheres and the case of $S^3$.

Findings

Manifolds with trivial $H_1(M;\mathbb{Z}_2)$ are connected sums of 8 $ ext{Z}_2$-homology spheres.

Irreducible manifolds with such actions are homeomorphic to $S^3$.

Extension of results to rational homology spheres with $H_1(M;\mathbb{Z}_2) \cong \text{Z}_2$ under fixed point assumptions.

Abstract

As a generalization of Davis-Januszkiewicz theory, there is an essential link between locally standard $(\Z_2)^n$-actions (or $T^n$-actions) actions and nice manifolds with corners, so that a class of nicely behaved equivariant cut-and-paste operations on locally standard actions can be carried out in step on nice manifolds with corners. Based upon this, we investigate what kinds of closed manifolds admit locally standard $(\Z_2)^n$-actions; especially for the 3-dimensional case. Suppose $M$ is an orientable closed connected 3-manifold. When $H_1(M;\Z_2)=0$, it is shown that $M$ admits a locally standard $(\Z_2)^3$-action if and only if $M$ is homeomorphic to a connected sum of 8 copies of some $\Z_2$-homology sphere $N$, and if further assuming $M$ is irreducible, then $M$ must be homeomorphic to $S^3$. In addition, the argument is extended to rational homology 3-sphere $M$ with $H_1(M;\Z_2) \cong \Z_2$ and an additional assumption that the $(\Z_2)^3$-action has a fixed point.