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Register a new userOn 3-manifolds with locally-standard (Z_2)^3-actions
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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.
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We show that the classical example $X$ of a 3-dimensional generalized manifold constructed by van Kampen is another example of not homologically locally connected (i.e. not HLC) space. This space $X$ is not locally homeomorphic to any of the compact metrizable 3-dimensional manifolds constructed in our earlier paper which are not HLC spaces either.
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In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard $(mathbb{Z}_2)^3$-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from $mathbb{R}P^3$ and $S^1timesmathbb{R}P^2$ with certain $(mathbb{Z}_2)^3$-actions under these six operations. As an application, we classify all 3-dimensional small covers up to $({Bbb Z}_2)^3$-equivariant unoriented cobordism.
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