Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $

Question

How to show, that an inclusion of homogenious spaces $$O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $$ is homotopy equivalence? The big space is the space of complex structures on $\mathbb{R}^{2n}$. This is an exercise in the book of McDuff and Salamon on symplectic topology, and they recommend to use polar decomposition, but it gives a deformational retraction $GL(2n,\mathbb{R}) \to O(2n)$, but does not give a map of homogenious spaces, since it is not $GL(n,\mathbb{C})$-equivariant.

Answer 1

You could probably write down an explicit homotopy inverse, but here is a quick way of seeing this:

We have fiber bundles $O(2n)/U(n) \rightarrow BU(n) \rightarrow BO(2n)$ and $GL_{2n}(\mathbb{R})/GL_{n}(\mathbb{C}) \rightarrow BGL_{n}\mathbb{C} \rightarrow BGL_{2n}\mathbb{R}$ and a map from the first to the second which induces homotopy equivalences on the base and total space, whence the fibers are homotopy equivalent.