Cobordism ring

Summary

The cobordism ring is the graded ring whose elements in degree $n$ are cobordism classes of closed smooth $n$-manifolds. Its addition is disjoint union and its multiplication is cartesian product. There are versions for unoriented, oriented, complex and other structured manifolds. Thom computed the unoriented ring and showed that cobordism groups are homotopy groups of Thom spectra. This brought homotopy theory into the classification of manifolds.