Spaces, Bundles and Characteristic Classes in Differential Geometry

Abstract

Part II prepares the reader to see how some of the basic notions of differential geometry pass into non-commutative geometry. The basic notions presented in the first chapter are reconsidered in the second chapter from a non-commutative geometry view point. Differential geometry begins with the algebra \(\mathcal {A} = C^{\infty }(M)\) of smooth functions and builds up by adding multiple structures; classical index theory uses most of these structures. Non-commutative geometry is abstract index theory; its axioms comprise many of these structures. While differential geometry is built by summing up different structures, non-commutative geometry reverses this process. In differential geometry the commutativity and locality assumptions are built in by means of the construction of differential forms. There are two basic differences which summarise the passage from differential geometry to non-commutative geometry: in differential geometry (1) the basic algebra \(\mathcal {A} = C^{\infty }\)is commutative, has true derivations (differential fields), and has a topology—the Frechet topology; in non-commutative geometry, the basic algebra \(\mathcal {A}\)is not required to be commutative nor to have a topology, nor to have derivations, (2) in differential geometry, the basic algebra \(\mathcal {A}\) is used to produce local objects; in non-commutative geometry the locality assumption is removed. Non-commutative geometry finds and uses the minimal structure which stays at the foundation of geometry: of differential forms, product of (some) distributions, bundles, characteristic classes, cohomology/homology and index theory. The consequences of this discovery are far reaching.