$Z_2$-index of the grassmanian $G_{2n}^n$

TL;DR

This paper investigates the $Z_2$-index of the real Grassmannian $G_{2n}^n$, providing bounds and exact values for certain cases, revealing topological properties related to orthogonal complements.

Contribution

It establishes the homological index of the $Z_2$ action on $G_{2n}^n$, especially proving it equals $2n-1$ when $n$ is a power of two.

Findings

Homological index bounds for the $Z_2$ action on $G_{2n}^n$

Exact index value of $2n-1$ when $n$ is a power of two

Topological insights into the structure of Grassmannians under orthogonal complement action

Abstract

We study the real Grassmann manifold $G_{2n}^n$ (of $n$-subspaces in $\mathbb R^{2n}$), and the action of $Z_2$ on it by taking the orthogonal complement. The homological index of this action is estimated from above and from below. In case $n$ is a power of two it is shown that $\hind G_{2n}^n=2n-1$.