Theorem 16.7.
If \(X\) is paracompact (e.g. compact Hausdorff), \(f, g \colon X \to Y\) are homotopic, and \(E \to Y\) is a vector bundle, then \(f^*E \cong g^*E\text{.}\)
Section 16.3 Homotopy Invariance
A cohomology theory must be homotopy invariant, so the first test for \(\mathrm{Vect}_k\) is whether homotopic maps induce the same pullback. This holds as soon as the domain is paracompact.
We only sketch the proof, which is Theorem 1.6 in Hatcher, Vector Bundles and K-Theory. Let \(F \colon X \times I \to Y\) be a homotopy from \(f\) to \(g\text{.}\) Then \(f^*E\) and \(g^*E\) are the restrictions of the single bundle \(F^*E\) over \(X \times I\) to the two ends. So it suffices to show that for every vector bundle \(E'\) over \(X \times I\text{,}\) the restrictions over \(X \times \{0\}\) and \(X \times \{1\}\) are isomorphic (Hatcher, Prop. 1.7). Compactness of \(I\) gives an open cover \(\{U_i\}\) of \(X\) such that \(E'\) is trivial over each \(U_i \times I\text{,}\) and by paracompactness we may take it countable with a subordinate, locally finite partition of unity \(\{\varphi_i\}\) (Hatcher, Lemma 1.21). Note that isomorphisms cannot simply be averaged with the \(\varphi_i\text{,}\) since an average of invertible maps can be singular. Instead we move from the bottom to the top of \(X \times I\) in steps. Let \(\psi_i = \varphi_1 + \dots + \varphi_i\text{,}\) so \(\psi_0 = 0\text{,}\) and let \(E_i\) be the restriction of \(E'\) to the graph of \(\psi_i\text{.}\) The graphs of \(\psi_i\) and \(\psi_{i-1}\) differ only over \(U_i\text{,}\) where \(E'\) is trivial, so the trivialization gives an isomorphism \(h_i \colon E_i \to E_{i-1}\) which is the identity outside \(U_i\text{.}\) By local finiteness, near each point only finitely many \(h_i\) differ from the identity, so the infinite composite \(h_1 h_2 h_3 \cdots\) is a well-defined isomorphism from \(E'|_{X \times \{1\}}\) to \(E'|_{X \times \{0\}}\text{.}\)
Corollary 16.8.
A homotopy equivalence \(f \colon X \to Y\) of paracompact spaces induces a bijection
\begin{equation*}
f^* \colon \mathrm{Vect}_k(Y) \xrightarrow{\;\cong\;} \mathrm{Vect}_k(X), \qquad [E] \mapsto [f^*E].
\end{equation*}
Proof.
Let \(g \colon Y \to X\) be a homotopy inverse. Since \(X\) is paracompact and \(gf \simeq \mathrm{id}_X\text{,}\) Theorem 16.7 gives \(f^*g^* = (gf)^* = \mathrm{id}\) on \(\mathrm{Vect}_k(X)\text{.}\) Similarly \(g^*f^* = \mathrm{id}\) on \(\mathrm{Vect}_k(Y)\text{,}\) since \(Y\) is paracompact.
In particular every vector bundle over a contractible paracompact space is trivial.
Remark 16.9.
In the next chapter we will see that, over a CW complex \(X\text{,}\) isomorphism classes of rank-\(k\) complex vector bundles are in natural bijection with homotopy classes of maps \(X \to \mathrm{Gr}_k(\mathbb{C}^\infty)\) (and similarly for real bundles using \(\mathrm{Gr}_k(\mathbb{R}^\infty)\)). The Grassmannians are thus classifying spaces for vector bundles.
