Thom spectrum

Summary

A Thom spectrum is a spectrum whose spaces are the Thom spaces of a compatible sequence of vector bundles. The basic examples are $MO$, $MSO$ and $MU$, built from the universal bundles over the classifying spaces $BO(n)$, $BSO(n)$ and $BU(n)$. By the Pontryagin–Thom construction, the homotopy groups of these spectra are the cobordism groups of manifolds with the matching structure, as in Thom's cobordism theorem. So Thom spectra turn geometric classification problems into problems of stable homotopy theory.