K-Theory

The topological index (see Todd class, d operator, Chern character from [1], [2], [3], [4]) is expressed via K-theory of vector bundles. This older topological K-theory has been virtually complete since the 70s (see e.g. [5]).

However, the algebraic version of K-theory is more general and extensible. By the Serre-Swan correspondence between vector bundles and projective modules it coincides with topological K-theory in the commutative case, thus pointing the way to a 'noncommutative topology'; cf. [6], [7].
 
 
References:

[1]  Palais: Seminar on the Atiyah-Singer index theorem

[2]  Atiyah & Hirzebruch: Analytic cycles on complex manifolds. Topology 1 (1962), §3

[3]  Hirzebruch: Topological methods in algebraic geometry

[4]  Kobayashi/Nomizu: Foundations of differential geometry, Chapter XII

[5]  Karoubi: K-Theory

[6]  Wegge-Olsen: K-Theory and C*-Algebras

[7]  Gracia-Bondia, Figueroa, Varilly: Elements of Noncommutative Geometry