Small Cover and Halperin-Carlsson Conjecture -II

TL;DR

This paper characterizes when the total Z_2-Betti number sum of certain principal bundles over small covers equals 2^m, showing it occurs precisely when the bundle is a product of spheres and the base is a generalized real Bott manifold.

Contribution

It proves an exact condition for the Betti number sum to equal 2^m, linking topological structure to the geometry of small covers and principal bundles.

Findings

Total Betti number sum equals 2^m iff the bundle is a product of spheres.

Base manifold must be a generalized real Bott manifold.

Connectedness of M^n is essential for the characterization.

Abstract

For a small cover Q^n and any principal (Z_2)^m-bundle M^n over Q^n, it was shown in a previous work of the author that the total sum of Z_2-Betti numbers of M^n is at least 2^m. In this paper, we prove that when M^n is connected, the total sum of Z_2-Betti numbers of such an M^n exactly equals 2^m if and only if M^n is homeomorphic to a product of spheres, and Q^n in this case must be a generalized real Bott manifold (or equivalent, Q^n is a small cover over a product of simplices).