A question in the latest differential geometry assignment asks us to show that every vector bundle over $[0,1]$ is trivial. However, the proofs available online typically use machinery we haven't been exposed to yet, like affine connections, or Cech cohomology. So I've been thinking about what elementary arguments can be used to show this. I envision that one possible proof would look a bit like this:
- $[0,1]$ is homotopy equivalent to the point.
- Every vector bundle over the point is trivial.
- Hence every vector bundle over $[0,1]$ is trivial.
Step 3 is the part I'm unsure about. What I'd like to know is
Question. If two topological manifolds are homotopy equivalent, what does this tell us about the categories of vector bundles over them?
Isomorphism classes of vector bundles of rank $r$ on $X$ are classifed by homotopy classes of maps $f : X \to \text{Gr}(r, \infty)$, this is the theorem 1.16 in the notes by Hatcher. (Here $\text{Gr}(r, \infty)$ is the infinite grassmanian of $r$-planes).