Topological types of 3-dimensional small covers

TL;DR

This paper classifies 3-dimensional small covers, which are manifolds with specific group actions, by introducing six operations and showing how all such covers can be constructed from basic examples, leading to a cobordism classification.

Contribution

It introduces six combinatorial operations on 3D small covers and demonstrates their sufficiency to generate all such covers from basic manifolds, enabling classification.

Findings

All 3D small covers can be obtained from $ ext{RP}^3$ and $S^1 imes ext{RP}^2$ using six operations.

The paper provides a classification of these covers up to equivariant unoriented cobordism.

The six operations are characterized by their combinatorial nature and their role in the construction process.

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