How do we construct an associated bundle $V_{n, q}(\omega)$ over $B$ with typical fiber $V_{n - q}(F)$?

Question

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + 1} V_{n - q}(\mathbb{C}^n) \cong \mathbb{Z}.$$Given a complex $n$-plane bundle $\omega$ over a CW-complex $B$ with typical fiber $F$, how do we construct an associated bundle $V_{n- q}(\omega)$ over $B$ with typical fiber $V_{n - q}(F)$?

Progress. So my idea is to consider the vector bundle $\text{Hom}(B \times \mathbb{C}^{n-q}, \omega)$ over $B$, and take the open subvariety of homomorphisms $u$ such that $u_b$ is injective for each $b \in B$. But I am not sure as to whether this would work or not... Could anybody help?

Answer 1

One way to do (i.e. define, convince yourself they exist etc) these "associated bundle" constructions is to take an open cover $\{U_\alpha\}$ of the base $B$ such that your bundle $\omega$ is trivial over each $U_\alpha$, do the construction locally, then reassemble. Concretely, fix a trivialisation $$\phi_\alpha : \omega|_{U_\alpha} \rightarrow U_\alpha \times F$$ for each $\alpha$, and let the transition functions be $\psi_{\alpha\beta}$. Then take the local models $U_\alpha \times V_{n-q}(F)$ and glue together using the transition functions induced by the $\psi_{\alpha \beta}$.

An alternative construction in your case is to take the Whitney sum of $n-q$ copies of $\omega$ then take the subspace consisting of orthonormal $(n-q)$-tuples of vectors. This is very similar to what you're suggesting. In fact you could modify your suggestion by restricting to the subspace of homomorphisms which preserve the inner product and recover exactly this.

There may be a more unified category-theoretic viewpoint to this.

Answer 2

Given a complex $n$-bundle $E \to B$ with generic fiber $F$, we construct a $V_{n - q}(\mathbb{C}^n)$ bundle $V(E)$ with generic fiber $V_{n - q}(F)$ by, of course, taking the fiber over a point whose original fiber was $F$ to be the set of complex $(n - q)$-hyperplanes in $F$, and defining the topology via local trivializations defined as follows. If $\mathbb{C}^n \times U \cong E|_U$ is a local trivialization in a neighborhood $U$ of the basepoint corresponding to $F$, this induces a map of abstract sets, from the moduli interpretation:$$V_{n - q}(\mathbb{C}^n) \times U \cong V(\mathbb{C}^n \times U) \cong V(E)|_U,$$which gives the topological structure of the bundle, since we know what said structure of the thing on the left is, obviously gives the same topology on overlaps, and of course gives the right topology when restricted to each fiber.