Gröbner bases in the mod $2$ cohomology of oriented Grassmann manifolds ${\tilde{G}}_{2^t,3}$

Abstract

For $n$ a power of two, we give a complete description of the cohomology algebra $H^{\ast }({\tilde{G}}_{n,3};{\mathbb{Z}}_2)$ of the Grassmann manifold ${\tilde{G}}_{n,3}$ of oriented $3$-planes in ${\mathbb{R}}^n$. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for $H^{\ast }({\tilde{G}}_{n,3};{\mathbb{Z}}_2)$.