Combinatorial constructions of three-dimensional small covers

TL;DR

This paper develops methods to construct all 3-dimensional small covers from basic manifolds using combinatorial operations on colored polytopes, extending previous results.

Contribution

It introduces a construction framework for 3D small covers via connected sum and surgery operations, generalizing prior work.

Findings

All 3D small covers can be built from T^3, RP^3, and S^1×RP^2.

The construction uses combinatorial operations on colored polytopes.

The approach generalizes and improves previous results by Lü-Yu.

Abstract

A small cover was introduced by Davis and Januszkiewicz as an $n$-dimensional closed manifold with a locally standard $Z_2)^n$-action such that its orbit space is a simple convex polytope. There exist a one-to-one correspondence between small covers and $(Z_2)^n$-colored polytopes. In this paper we study a construction of 3-dimensional small covers by using two operations called a connected sum and a surgery. These operations correspondent to combinatorial operations on $(Z_2)^3$-colored simple convex polytopes. We shall show that each 3-dimensional small cover can be constructed from $T^3$, $RP^3$ and $S^1 \times RP^2$ with two different $(Z_2)^3$-actions by using these operations. This result is a generalization and an improvement of L\"{u}-Yu's result.