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 \((\Z_2)^n\)-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.