The Cohomology Algebra of Polyhedral Product Objects

TL;DR

This paper computes the cohomology algebra of polyhedral product objects using diagonal tensor products and explores their applications in commutative algebra and homotopy theory.

Contribution

It introduces a unified method for computing cohomology of polyhedral products and links their homotopy types to character coproducts.

Findings

Cohomology algebra computed via diagonal tensor product.

Polyhedral product method applied to commutative algebra.

Homotopy types depend on character coproducts.

Abstract

In this paper, we compute the homology group and cohomology algebra of various polyhedral product objects uniformly from the point of view of diagonal tensor product. As applications, we introduce the polyhedral product method into commutative algebra and show that the homotopy types of polyhedral product spaces depend on not only the homotopy type of each summand pair but also on the character coproduct of the pair.