Thom space

Summary

The Thom space of a real vector bundle $E \to B$ is its unit disc bundle with the unit sphere bundle collapsed to a point. For compact $B$ it is the one-point compactification of the total space $E$. The Thom isomorphism identifies its reduced cohomology with the cohomology of $B$ shifted by the rank. Thom spaces of universal bundles form the Thom spectra that compute cobordism groups.