noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
Michael Atiyah and Isadore Singer associated two differently defined numbers to an elliptic operator on a manifold: the topological index and the analytical index. The index theorem asserts that the two are equal.
The original articles are
Michael Atiyah, Isadore Singer, The index of elliptic operators I, Ann. of Math. (2) 87 (1968) pp. 484–530; III, Ann. of Math. (2) 87 (1968) pp. 546–604; IV, Ann. of Math. (2) 93 (1971) pp. 119–138; V, Ann. of Math. (2) 93 (1971) pp. 139–149
Michael Atiyah, Graeme Segal, The index of elliptic operators II, Ann. of Math. (2) 87 (1968) pp. 531–545
A proof of the Atiyah-Singer index theorem in terms of KK-theory/E-theory has been given by Nigel Higson, an account is in
A lighning review of the proof is on the last pages of
See also
The index theorem is supposed to have an interpretation in terms of the quantum field theory of the superparticle on the given space.
Traditional physics arguments along these lines include for instance
P. Windey, Supersymmetric quantum mechanics and the Atiyah–Singer index theorem, Acta Physica Polonica, B15 (1984). (PDF)
Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (EUCLID)