Cohomological rigidity and the number of homeomorphism types for small covers over prisms

TL;DR

This paper classifies all small covers over 3D prisms up to homeomorphism using cohomological invariants, proving cohomological rigidity and calculating the number of distinct homeomorphism types.

Contribution

It introduces a complete invariant system for classifying small covers over prisms and proves cohomological rigidity for these manifolds.

Findings

Cohomological invariants fully classify small covers over prisms in most cases.

Cohomological rigidity holds for all small covers over a prism.

Number of homeomorphism types over a prism is explicitly calculated.

Abstract

In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) \cite[Chapter 8, \S 2 Gluing Manifolds Together]{h}, we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimensional prism $P^3(m)$ with $m\geq 3$. We introduce two invariants from colored prisms and other two invariants from ordinary cohomology rings with ${\Bbb Z}_2$-coefficients of small covers. These invariants can form a complete invariant system of homeomorphism types of all small covers over a prism in most cases. Then we show that the cohomological rigidity holds for all small covers over a prism $P^3(m)$ (i.e., cohomology rings with ${\Bbb Z}_2$-coefficients of all small covers over a $P^3(m)$ determine their homeomorphism types). In addition, we also calculate the number of homeomorphism types of all small covers over $P^3(m)$.