Cohomology rings of oriented Grassmann manifolds $\widetilde G_{2^t,4}$

TL;DR

This paper describes the mod 2 cohomology algebra of oriented Grassmann manifolds $ ilde G_{2^t,4}$, providing a presentation as a quotient of a polynomial algebra, identifying a Gr"obner basis, and establishing an additive basis.

Contribution

It offers a new explicit algebraic description and Gr"obner basis for the cohomology of $ ilde G_{2^t,4}$, advancing understanding of their topological structure.

Findings

Explicit presentation of the cohomology algebra as a quotient

Identification of a Gr"obner basis for the ideal

Construction of an additive basis for cohomology

Abstract

We give a description of the mod 2 cohomology algebra of the oriented Grassmann manifold $\widetilde G_{2^t,4}$ as the quotient of a polynomial algebra by a certain ideal. In the process we find a Gr\"obner basis for that ideal, which we then use to exhibit an additive basis for $H^*(\widetilde G_{2^t,4};\mathbb Z_2)$.