On the cohomology ring and upper characteristic rank of Grassmannian of oriented $3$-planes

TL;DR

This paper investigates the mod 2 cohomology ring of the Grassmannian of oriented 3-planes, providing degree information of indecomposables, bounds on cup length, and insights into the characteristic rank.

Contribution

It offers a partial description of the cohomology ring of oriented 3-plane Grassmannians, including bounds on cup length and the equality of upper characteristic rank with that of the tautological bundle.

Findings

Determined degrees of indecomposable elements in the cohomology ring.

Provided bounds on the cup length of the Grassmannian.

Showed the upper characteristic rank equals the characteristic rank of the tautological bundle.

Abstract

In this paper we study the mod $2$ cohomology ring of the Grasmannian $\widetilde{G}_{n,3}$ of oriented $3$-planes in $\mathbb{R}^n$. We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This partial description allows us to provide lower and upper bounds on the cup length of $\widetilde{G}_{n,3}$. As another application, we show that the upper characteristic rank of $\widetilde{G}_{n,3}$ equals the characteristic rank of $\widetilde{\gamma}_{n,3}$, the oriented tautological bundle over $\widetilde{G}_{n,3}$.