Small covers as pullbacks from the simplex

TL;DR

This paper characterizes small covers that are pullbacks from the simplex, linking their topological properties to cohomological conditions and Betti number relations.

Contribution

It introduces and characterizes a new class of small covers derived from the simplex, with multiple equivalent topological and cohomological conditions.

Findings

Equivalent conditions for small covers as pullbacks from the simplex.

Torsion-freeness of odd-degree integral cohomology.

Vanishing of the first Steenrod square on even-degree mod 2 cohomology.

Abstract

We introduce and study small covers that are pullbacks from the simplex, extending pullbacks from the linear model. Our main result gives several equivalent characterizations of this class, including torsion-freeness of odd-degree integral cohomology, vanishing of the first Steenrod square on even-degree mod $2$ cohomology, and relations among integral and mod $2$ Betti numbers.