Small Cover and Halperin-Carlsson Conjecture

TL;DR

This paper proves the Halperin-Carlsson conjecture for free (Z_2)^m actions on compact manifolds with small cover orbit spaces and characterizes certain principal bundles over small covers.

Contribution

It establishes the conjecture for a new class of manifolds and describes the structure of principal (Z_2)^m bundles over small covers.

Findings

Halperin-Carlsson conjecture holds for small cover orbit spaces.

Connected principal (Z_2)^m bundles over small covers are partial quotients of real moment-angle manifolds.

Provides structural insights into (Z_2)^m actions on manifolds.

Abstract

We prove that the Halperin-Carlsson conjecture holds for any free (Z_2)^m action on a compact manifold whose orbit space is a small cover. In addition, we show that if the total space of a principal (Z_2)^m bundle over a small cover is connected, it must be equivalent to a partial quotient of the corresponding real moment-angle manifold with some canonical Z_2-torus action.