Homotopy equivalence between the Grassmannian Gr_{n,m} and Gr_n \times Gr_m. The following assertion appears in a paper I am reading, and I can't seem to verify it.
Let $\text{Gr}_{n,m}$ denote the set of pairs $(V,W)$ where $V$ and $W$ are as follows.


*

*$V$ is an $n$-dimensional subspace of $\mathbb{C}^{\infty}$.

*$W$ is an $m$-dimensional subspace of $\mathbb{C}^{\infty}$.

*$V$ and $W$ are orthogonal.


The space $\text{Gr}_{n,m}$ has an obvious topology.  If $\text{Gr}_n$ and $\text{Gr}_m$ are the usual Grassmannians of $n$ and $m$ planes in $\mathbb{C}^{\infty}$, then there is an obvious map $\psi : \text{Gr}_{n,m} \rightarrow \text{Gr}_n \times \text{Gr}_m$.
The map $\psi$ is almost a homeomorphism, but not quite because of condition 3 above.  The paper claims that $\psi$ is a homotopy equivalence.
Thanks for any help!
 A: Look at the canonical principal $U(n)\times U(m)$ bundle over $Gr_{n,m}$  given by pairs of orthonormal frames $(v_1,\ldots, v_n), (w_1,\ldots w_m)$. Its total space is the set of all orthonormal $n+m$-frames in $\mathbb C^\infty$. It's contractible (that's well-known) and hence $Gr_{n,m}$ is a homotopy $B_{U(n)\times U(m)}$ which is clearly homotopy equivalent to $B_{U(n)}\times B_{U(m)}=Gr_n\times Gr_m$. To see that the natural map $Gr_{n,m}\to Gr_n\times Gr_m$ is the one inducing an equivalence notice that it's obviously covered by a map of principal bundles and hence the result follows by 5-lemma since total spaces are contractible and the fibers are the same.
A: The forgetful map $Gr_{n,m} \to Gr_n$ that drops $W$ is a fiber bundle (exercise), and the map $Gr_{n,m} \to Gr_n \times Gr_m$ is a map of fiber bundles.  It's an equivalence on the (connected) base space, so it suffices to check that the map of fibers is an equivalence.
The fibers over $V$ are, respectively: $m$-dimensional subspaces in $V^\perp \subset \mathbb{C}^\infty$, and $m$-dimensional subspaces in $\mathbb{C}^\infty$.
The inclusion of one infinite-dimensional complex vector space in another induces a homotopy equivalence of Grassmannians; you could construct an explicit homotopy equivalence by choosing an appropriate basis, or you could argue that the associated map of Stiefel manifolds is a homotopy equivalence (both are contractible, so this is easy) and so it passes to an equivalence after taking the quotient by the general linear group.
