Topological complexity of oriented Grassmann manifolds

TL;DR

This paper investigates the $ ext{Z}_2$-zero-divisor cup-length of oriented Grassmann manifolds $ ilde{G}_{n,3}$, providing bounds and exact values for many cases, thereby establishing lower bounds for their topological complexity.

Contribution

It offers new bounds and exact calculations for the $ ext{Z}_2$-zero-divisor cup-length of $ ilde{G}_{n,3}$, advancing understanding of their topological complexity.

Findings

Exact values computed for infinitely many $n$

Bounds differ by 1 in many cases

Lower bounds for topological complexity established

Abstract

We study the $\mathbb Z_2$-zero-divisor cup-length, denoted by $\operatorname{zcl}_{\mathbb Z_2}(\widetilde G_{n,3})$, of the Grassmann manifolds $\widetilde G_{n,3}$ of oriented $3$-dimensional vector subspaces in $\mathbb R^n$. Some lower and upper bounds for this invariant are obtained for all integers $n\ge6$. For infinitely many of them the exact value of $\operatorname{zcl}_{\mathbb Z_2}(\widetilde G_{n,3})$ is computed, and in the rest of the cases these bounds differ by 1. We thus establish lower bounds for the topological complexity of Grassmannians $\widetilde G_{n,3}$.