Corollary 9.2.16. Let \(X\) be a paracompact Hausdorff space. There is a natural bijection \[ \left [X,\bigsqcup _{k=0}^{\infty }\Gr _k(\C ^\infty )\right ] \xrightarrow {\cong } \pi _0 \Vect (X). \] If \(X\) is a cell complex, this bijection can equivalently be written as \[ \pi _0\Hom _{\An }\left (\Pi _{\infty }X,\bigsqcup _{k=0}^{\infty }\bB U(k)\right ) \xrightarrow {\cong } \pi _0\Vect (X). \]
Proof. Both sides of the first bijection decompose according to locally constant functions \(r\colon X\to \N \). For such a function, let \(X_k:=r^{-1}(k)\). These subspaces are clopen, and hence paracompact Hausdorff, and they form a topological disjoint-union decomposition \[ X=\bigsqcup _{k\geq 0}X_k. \] A continuous map \(X\to \bigsqcup _{k\geq 0}\Gr _k(\C ^\infty )\) with component function \(r\) is precisely a family of continuous maps \(X_k\to \Gr _k(\C ^\infty )\). Similarly, a vector bundle with rank function \(r\) is precisely a family of rank \(k\) vector bundles over the \(X_k\). The first bijection therefore follows on each clopen stratum from Theorem 9.2.4.
The Schubert cell structures above also make \(\bigsqcup _{k\geq 0}\Gr _k(\C ^\infty )\) a cell complex. If \(X\) is a cell complex as well, then Corollary 2.3.7, Corollary 9.2.15 identifies the source of the first bijection with the source of the second. □
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