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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2008
- DOI:
- arXiv:
- arXiv:0807.3062
- Bibcode:
- 2008arXiv0807.3062L
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Algebraic Topology;
- Mathematics - Combinatorics;
- 57M60;
- 57M50;
- 57S17
- E-Print:
- 17 pages, 10 figures, significant expansions are made to the previous version, and some examples and figures are added