Noncommutative topology - homotopy functors and E-theory
Noncommutative topology - homotopy functors and E-theory
- Authors: Kasper K.S. Andersen and Jesper Grodal
- Date: Feb 1997.
- Status: Accepted.
- Pages: 30.
- You can get a copy of our project by clicking below:
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.