History
Cobordism had its roots in the (failed) attempt by Henri Poincaré in 1895 to define homology purely in terms of manifolds (Dieudonné 1989, p. 289). Poincaré simultaneously defined both homology and cobordism, which are not the same, in general. See Cobordism as an extraordinary cohomology theory for the relationship between bordism and homology.
Bordism was explicitly introduced by Lev Pontryagin in geometric work on manifolds. It came to prominence when René Thom showed that cobordism groups could be computed by means of homotopy theory, via the Thom complex construction. Cobordism theory became part of the apparatus of extraordinary cohomology theory, alongside K-theory. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the Hirzebruch–Riemann–Roch theorem, and in the first proofs of the Atiyah–Singer index theorem.
In the 1980s the category with compact manifolds as objects and cobordisms between these as morphisms played a basic role in the Atiyah–Segal axioms for topological quantum field theory, which is an important part of quantum topology.
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