From Differential Geometry to Non-commutative Geometry and Topology by Neculai S. Teleman

Author:Neculai S. Teleman
Language: eng
Format: epub, pdf
ISBN: 9783030284336
Publisher: Springer International Publishing

Define .

One has the exact sequence

(3.43)

One has the analogue of Proposition 3.5.

Proposition 3.10

The sub-complex has trivial homology in the complex .

Proof

One has the identity on

(3.44)

Any element of is the -multiple of an element of C ∗. Let be mapping given by the division by . Then satisfies the identity

(3.45)

More applications of the techniques presented in this section can be found in [113].

3.4 Combinatorics Behind Homology Theories

Here we present the combinatorics which stays at the foundations of some of the most familiar homology theories (non-localised Alexander–Spanier homology and cohomology [45], Hochschild [68, 82, 99] and modified Hochschild homology [119, 123]) (later on called topological Hochschild homology) in the cases in which the product is pointwisely defined, as in the case of smooth functions. This discussion does not regard the case of the algebra of integral or pseudo-differential operators. The Hochschild and modified Hochschild homology are built on associative algebras, possibly endowed with a topological structure. The construction we present here extends the definition of Hochschild and modified Hochschild homology to a mixed context in which the role of the topological tensor products is replaced by a functor which associates with any power of a base space a set of functions with certain regularity properties, as it occurs in the case of smooth functions. This situation is related to the case discussed in Sect. 3.1. For more details the reader may consult [120]. More related information may be found in [119, 123].

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