Pontryagin–Thom construction

Summary

The Pontryagin–Thom construction starts from a closed submanifold with a tubular neighbourhood. It gives a map from the ambient manifold to the Thom space of the normal bundle. It collapses everything outside the tubular neighbourhood to a point. Pontryagin used it to identify framed cobordism classes of submanifolds of spheres with homotopy groups of spheres. Thom used it to compute cobordism groups as homotopy groups of Thom spaces, see Thom's cobordism theorem.