Spaces, Bundles and Characteristic Classes in Differential Geometry

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 A = C ∞ ( M ) $$\mathcal {A} = C^{\infty }(M)$$ of smooth functions Smooth functions and builds up by adding multiple structures; classical index theory uses most of these structures. Non-commutative geometry is abstract index theory 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 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 A = C ∞ $$\mathcal {A} = C^{\infty }$$ is commutative, has true derivations (differential fields) Differential fields , and has a topology—the Fréchet topology; in non-commutative geometry, the basic algebra A $$\mathcal {A}$$ is not required to be commutative nor to have a topology, nor to have derivations, (2) in differential geometry, the basic algebra A $$\mathcal {A}$$ 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.