Computing Cohomology Rings in Cubical Agda

TL;DR

This paper presents the first mechanized definition of cohomology rings within Homotopy Type Theory using Cubical Agda, enabling concrete computations of cohomology rings for classical spaces.

Contribution

It introduces a formalization of cohomology rings as quotients of polynomial rings in Cubical Agda, extending previous work with a fully mechanized, constructive approach.

Findings

Successfully computed cohomology rings of spheres and Klein bottle

Defined cohomology rings as quotients of multivariate polynomial rings

Provided a constructive formalization suitable for concrete computations

Abstract

In Homotopy Type Theory, cohomology theories are studied synthetically using higher inductive types and univalence. This paper extends previous developments by providing the first fully mechanized definition of cohomology rings. These rings may be defined as direct sums of cohomology groups together with a multiplication induced by the cup product, and can in many cases be characterized as quotients of multivariate polynomial rings. To this end, we introduce appropriate definitions of direct sums and graded rings, which we then use to define both cohomology rings and multivariate polynomial rings. Using this, we compute the cohomology rings of some classical spaces, such as the spheres and the Klein bottle. The formalization is constructive so that it can be used to do concrete computations, and it relies on the Cubical Agda system which natively supports higher inductive types and computational univalence.