Editing Atiyah–Singer index theorem (section)

===Cobordism===
The initial proof was based on that of the [[Hirzebruch–Riemann–Roch theorem]] (1954), and involved [[cobordism theory]] and [[pseudo-differential operator|pseudodifferential operator]]s.

The idea of this first proof is roughly as follows. Consider the ring generated by pairs (''X'', ''V'') where ''V'' is a smooth vector bundle on the compact smooth oriented manifold ''X'', with relations that  the sum and product of the ring on these generators are given by disjoint union and product of manifolds (with the obvious operations on the vector bundles), and any boundary of a manifold with vector bundle is 0. This is similar to the cobordism ring of oriented manifolds, except that the manifolds also have a vector bundle. The topological and analytical indices are both reinterpreted as functions from this ring to the integers. Then one checks that these two functions  are in fact both ring homomorphisms. In order to prove they are the same, it is then only necessary to check they are the same on a set of generators of this ring. Thom's cobordism theory gives a set of generators; for example, complex vector spaces with the trivial bundle together with certain bundles over even dimensional spheres. So the index theorem can be proved by checking it on these particularly simple cases.