Topological types of 3-dimensional small covers

Abstract

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 $({\mathbb{Z}}_2{)}^3$-equivariant unoriented cobordism.

34 pages with 40 figures, final version for publication

References in corpus (1)

• Graphs and $({\mathbb{Z}}_2{)}^k$-actions

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