Cup-length of oriented Grassmann manifolds via Gröbner bases

Abstract

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the ${\mathbb{Z}}_2$-cup-length of the Grassmann manifold ${\tilde{G}}_{n,3}$ of oriented $3$-planes in ${\mathbb{R}}^n$. Along the way, we calculate the heights of the Stiefel--Whitney classes of the canonical vector bundle over ${\tilde{G}}_{n,3}$.