Topological K-theory

Summary

Topological K-theory assigns to a compact Hausdorff space $X$ the group $K^0(X)$ of formal differences of complex vector bundles on $X$. It extends to a generalized cohomology theory $K^n(X)$, which is periodic of period $2$ by the Bott periodicity theorem. It was developed by Atiyah and Hirzebruch, following Grothendieck's work in algebraic geometry. It gives short proofs of results such as the Hopf invariant one theorem and is the natural home of the Atiyah–Singer index theorem.