Thom isomorphism in K-theory

Remarks

The class $\lambda_E$ is represented by a Koszul complex of exterior powers over $E$: $$0 \to \Lambda^0 p^*E \to \Lambda^1 p^*E \to \cdots \to \Lambda^n p^*E \to 0.$$ Its maps at a point $v \in E$ are $\omega \mapsto v \wedge \omega$. It is exact away from the zero section, so it defines a class with compact support. The proof starts with a trivial bundle over a point, where the statement is $\tilde{K}(S^{2n}) \cong \Z$ from Bott periodicity. It then glues over a finite cover of $X$ by trivializing open sets with Mayer–Vietoris sequences.

Example: for $X$ a point and $E = \C^n$, the theorem says $\tilde{K}(S^{2n}) \cong K(\text{point}) = \Z$.

The complex structure matters: a real vector bundle need not have a Thom class in K-theory. Real bundles with a K-theory Thom class are called K-oriented; they include all real bundles with a spin$^c$ structure. The theorem is the K-theory analogue of the Thom isomorphism in ordinary cohomology.

Going deeper

For a student

The K-theory Thom isomorphism is the key step in the construction of push-forward maps in K-theory, also called Gysin maps. Take an embedding of manifolds with a complex normal bundle. One goes from the submanifold to the Thom space of the normal bundle by the Thom isomorphism. Then one goes to the big manifold by collapsing the complement of a tubular neighbourhood. This is how the topological index in the Atiyah–Singer index theorem is defined. Comparing the K-theory Thom class with the cohomology Thom class through the Chern character produces the Todd class, which explains the Todd class in the Riemann–Roch formulas.