  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.
