Noncommutative topology - homotopy functors and E-theory

Noncommutative topology - homotopy functors and E-theory

Abstract: This is one of our Graduate projects from the University of Copenhagen. It is written under the supervision of assistant professor Søren Eilers.
The main goal of this `fagprojekt' is to explore the bifunctor E of Connes and Higson, generalizing the Kasparov bifunctor KK. This functor E has a concrete realization as homotopy classes of certain `almost *-homomorphisms' (asymptotic morphisms). We start by examining the properties of these asymptotic morphisms and go on to we establish conditions under which homotopy classes of asymptotic homomorphisms coincide with homotopy classes of *-homomorphisms.
In order to understand the role of the E bifunctor we are lead to examine the general properties of homotopy functors on the category of C*-algebras. We examine cofibrations in the category of C*-algebras, and introduce the notion of a `cofibration half exact' functor as being the natural generalization of the topological concept of an exact functor. We then show that a functor is half exact precisely if it is cofibration half exact and excisive. An excisive functor on the category of C*-algebras is defined to be one which does not distinguish between the kernel and the `homotopy kernel' (the mapping cone) of a surjective *-homomorphism.
We then proceed to introduce the Puppe sequence associated to a morphism of C*-algebras mimicking the analogous topological constructions. This allows us to establish the existence of long exact sequences corresponding to various functors. Finally, we use the obtained results obtained results to introduce the bifunctor E and establish its basic properties.
Most of the `fagprojekt', except for some of Section 3 on cofibrations in the category of C*-algebras, is an exposition of already known material.
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<jg@math.ku.dk>/April 8, 1997.