Complement of zero section homotopy equivalent to $\mathrm{Gr}_{k-1}(\mathbb{C}^{\infty})$

Question

Let $\mathrm{Gr}_{k}(\mathbb{C}^{\infty})$ be the Grassmanian of complex $k$-dimensional subspaces of $\mathbb{C}^{\infty}$, and let $\gamma:E\to \mathrm{Gr}_{k}(\mathbb{C}^{\infty})$ be the tautological $k$-plane bundle, whose fiber over a point $X\in \mathrm{Gr}_{k}(\mathbb{C}^{\infty})$ is the space $$ \gamma^{-1}(X)=\{v\in \mathbb{C}^{\infty}\mid v\in X\}. $$ Let $E_0\subset E$ be the complement of the zero section of $\gamma$, i.e. $$ E_0=\{(X,v)\in \mathrm{Gr}_{k}(\mathbb{C}^{\infty})\times \mathbb{C}^{\infty}\mid v\in X,\; v\neq 0\}. $$

Let $f:E_0\to \mathrm{Gr}_{k-1}(\mathbb{C}^{\infty})$ be the map $$ f(X,v)=X\cap v^{\perp}\in \mathrm{Gr}_{k-1}(\mathbb{C}^{\infty}), $$ where the orthogonal complement $v^{\perp}\subset \mathbb{C}^{\infty}$ is computed with respect to the standard Hermitian inner product on $\mathbb{C}^{\infty}$, $ v\cdot w=\sum_{i=1}^{\infty}v_i\bar{w}_i. $ In their computation of the cohomology ring of $\mathrm{Gr}_{k}(\mathbb{C}^{\infty})$, Milnor-Stasheff show (Theorem 14.5) that $f$ induces an isomorphism of the cohomology rings $$ f^*:H^*(\mathrm{Gr}_{k-1}(\mathbb{C}^{\infty});\mathbb{Z})\cong H^*(E_0;\mathbb{Z}). $$

Question: Is $f$ in fact a homotopy equivalence, and if so, how can this be shown?

In the case $k=1$, I am able to show that $E_0$ and $\mathrm{Gr}_{k-1}(\mathbb{C}^{\infty})$ are homotopy equivalent: indeed, $\mathrm{Gr}_{k-1}(\mathbb{C}^{\infty})=\{0\}$ is contractible, and hence homotopy equivalent to the unit sphere $S^{\infty}\subset \mathbb{C}^{\infty}$ (which is also contractible), and the map $E_0\to S^{\infty}$, defined by $$ (X,v)\mapsto \frac{v}{\|v\|}, $$ is a homotopy equivalence.

Answer 1

A cohomology equivalence of simply connected spaces is a homotopy equivalence. This follows from the universal coefficient theorem and homology Whitehead.

It is well known that complex Grassmannians are simply connected, so what remains to be shown is that the complement of the zero section is simply connected.

This complement is homotopy equivalent to the sphere bundle of the tautological bundle by normalizing, hence it suffices to show the sphere bundle is simply connected. By our earlier remarks, the base is simply connected and the fiber is as well (assuming $k>1$) since it is homotopy equivalent to a sphere of dimension $2k-1$. Hence, the total space is simply connected by the $LES$ of a fiber bundle.

So Milnor and Stasheff's work to show this is a cohomology equivalence implies that it is a homotopy equivalence.