The cohomology algebra of polyhedral product spaces

TL;DR

This paper computes the cohomology algebra of polyhedral product spaces, revealing that similar cohomology homomorphisms can correspond to non-isomorphic cohomology rings, thus highlighting subtle algebraic distinctions.

Contribution

It provides explicit calculations of the cohomology algebra for all homology split polyhedral product spaces and demonstrates non-isomorphic cohomology rings with identical induced homomorphisms.

Findings

Cohomology rings of polyhedral product spaces are explicitly computed.

Identifies cases where cohomology homomorphisms are identical but rings are not isomorphic.

Highlights subtle differences in algebraic structures of polyhedral product spaces.

Abstract

In this paper, we compute the cohomology ring of all homology split polyhedral product spaces and the cohomology algebra over a field of all polyhedral product spaces. As an application, we give two polyhedral product spaces such that all the cohomology homomorphisms induced by inclusion map are the same, but the cohomology rings of the two polyhedral product spaces are not isomorphic.