Serre–Swan theorem

Remarks

The key step is that every vector bundle $E$ on a compact Hausdorff space is a direct summand of a trivial bundle $X \times \mathbb{K}^N$. One builds the embedding from finitely many local trivializations and a partition of unity, and takes the complement to be the orthogonal complement for a metric on the fibres. Then $\Gamma(E)$ is a direct summand of $\Gamma(X \times \mathbb{K}^N) = C(X)^N$. Conversely, an idempotent matrix $e \in M_N(C(X))$ gives the bundle whose fibre at $x$ is the image of $e(x)$.

Example: on the $2$-sphere, the tangent bundle corresponds to the module of continuous vector fields, which is projective but not free. In K-theory the theorem gives $K^0(X) \cong K_0(C(X))$, linking topological K-theory with algebraic K-theory of rings.

The compactness is used to embed $E$ in a trivial bundle of finite rank. Jean-Pierre Serre proved the analogous algebraic statement in 1955: for a commutative ring $A$, vector bundles on the affine scheme $\mathrm{Spec}\, A$ correspond to finitely generated projective $A$-modules. Richard Swan proved the topological version in 1962.

Going deeper

For a student

The theorem is the reason projective modules are thought of as vector bundles, and why algebraic K-theory of rings is modelled on topological K-theory. Trivial bundles correspond to free modules, Whitney sums to direct sums, and tensor products of bundles to tensor products of modules. Questions such as Serre's problem on projective modules over polynomial rings come from this dictionary. In operator algebras, the theorem motivates treating projective modules over a noncommutative C*-algebra as vector bundles over a "noncommutative space".