Isomorphism of Grassmannians

Question

I want to prove that two CW complexes $\mathrm{Gr}_{n}(\mathbb{R}^{n+k})$ and $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ are isomorphic to one another. I'm pretty sure I can just show that the number of $r$-cells in each of the complexes are equal. The isomorphism should follow from this. However, I'm a little unsure how to do this. Any help would be appreciated. Thanks.

Answer 1

The map $f \colon \mathrm{Gr}_{n}(\mathbb{R}^{n+k}) \to \mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ given by $f(X) = X^\perp$ is a homeomorphism.