Gr\"obner bases in the mod $2$ cohomology of oriented Grassmann manifolds $\widetilde G_{2^t,3}$

TL;DR

This paper provides a complete description of the mod 2 cohomology algebra of oriented Grassmann manifolds  G_{n,3} for n a power of two, using Grf6bner bases to find generators and relations.

Contribution

It introduces a reduced Grf6bner basis for the ideal related to the cohomology algebra of  G_{n,3} and presents an explicit additive basis.

Findings

Explicit Grf6bner basis for the cohomology algebra

Complete description of the algebra structure for n a power of two

Constructed additive basis for the cohomology algebra

Abstract

For $n$ a power of two, we give a complete description of the cohomology algebra $H^*(\widetilde G_{n,3};\mathbb Z_2)$ of the Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-planes in $\mathbb R^n$. We do this by finding a reduced Gr\"obner basis for an ideal closely related to this cohomology algebra. Using this Gr\"obner basis we also present an additive basis for $H^*(\widetilde G_{n,3};\mathbb Z_2)$.