K-orientation

Summary

A K-orientation of a real vector bundle is a Thom class for the bundle in complex K-theory. It is the K-theory analogue of an orientation, which is a Thom class in ordinary integral cohomology. Complex vector bundles have canonical K-orientations, and so do real bundles with a spin$^c$ structure, by the Atiyah–Bott–Shapiro construction. A K-orientation gives a Thom isomorphism in K-theory and push-forward maps, which are the basis of index theory.