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

Abstract

As a generalization of Davis-Januszkiewicz theory, there is an essential link between locally standard $({\mathbb{Z}}_2{)}^n$-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 $({\mathbb{Z}}_2{)}^n$-actions; especially for the 3-dimensional case. Suppose $M$ is an orientable closed connected 3-manifold. When $H_1(M;{\mathbb{Z}}_2)=0$, it is shown that $M$ admits a locally standard $({\mathbb{Z}}_2{)}^3$-action if and only if $M$ is homeomorphic to a connected sum of 8 copies of some ${\mathbb{Z}}_2$-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 $H_1(M;{\mathbb{Z}}_2)\cong {\mathbb{Z}}_2$ and an additional assumption that the $({\mathbb{Z}}_2{)}^3$-action has a fixed point.

17 pages, 10 figures, significant expansions are made to the previous version, and some examples and figures are added

References in corpus (1)

• Topological types of 3-dimensional small covers