Grassmannians classify vector bundles

Proof sketch of Theorem 9.2.4. We describe the main ingredients of Hatcher’s argument. The key observation is that, for a rank \(k\) vector bundle \(p\colon E \to X\), the following two types of data are equivalent:

(1)

A pair \((f,\phi )\) consisting of a continuous map \(f\colon X \to \Gr _k(\C ^\infty )\) and an isomorphism \(\phi \colon E \cong f^*(E_{\univ })\);

(2)

A continuous map \(g\colon E \to \C ^\infty \) whose restriction to each fiber is a \(\C \)-linear injection.

Indeed, an isomorphism \(E \cong f^*(E_{\univ })\) gives such a map by projecting \(f^*(E_{\univ }) \subseteq X \times \C ^\infty \) to \(\C ^\infty \). Conversely, a fiberwise linear injection \(g\) determines \(f(x) := g(E_x) \subseteq \C ^\infty \), and the evident map \(E \to E_{\univ }\) identifies \(E\) with \(f^*(E_{\univ })\).

Surjectivity is therefore reduced to constructing a map \(g\colon E\to \C ^\infty \) which is linear and injective on each fiber. Starting from a trivializing cover, one uses the countable-cover lemma for paracompact spaces to obtain a countable trivializing cover \((U_i)_{i\geq 1}\) together with a locally finite subordinate partition of unity \((\rho _i)_{i\geq 1}\). Let \(g_i\colon E|_{U_i}\to \C ^k\) be the coordinate map of a trivialization. The maps \[ v\longmapsto \rho _i(p(v))g_i(v), \] extended by zero outside \(U_i\), assemble into a map \(E\to (\C ^k)^\infty \cong \C ^\infty \). Local finiteness ensures that this map is continuous and takes values in the algebraic direct sum, while the partition-of-unity condition makes it injective on every fiber. This is the only point at which paracompactness is used in an essential way.

For injectivity, suppose two classifying maps \(f_0,f_1\colon X \to \Gr _k(\C ^\infty )\) pull back \(E_{\univ }\) to isomorphic bundles. The corresponding fiberwise injections \(g_0,g_1\colon E \to \C ^\infty \) may first be homotoped so that their images land in the odd and even coordinates, respectively. The straight-line homotopy \((1-t)g_0+tg_1\) then remains fiberwise injective for all \(t\), and hence determines a homotopy from \(f_0\) to \(f_1\).

For the countable-cover lemma, see Hatcher (2003), Lemma 1.21; the remaining continuity details are part of the proof of Hatcher (2003), Theorem 1.16. □