  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 $\hind G_{2n}^n=2n-1$.
