Topological types of 3-dimensional small covers

Details Topological types of 3-dimensional small covers Topological types of 3-dimensional small covers

2007-10-24

arxiv.org/abs/0710.4496v4

Forum Math. 23 (2011), 245-284

Mathematics

Geometric Topology

34 pages with 40 figures, final version for publication

Scientific paper

10.1515/FORM.2011.008

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 $S^1\times\mathbb{R}P^2$ 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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