Connection in the KK-Theory

Question

I have some questions about the connection in the KK-Theory.

1)The definition is complicated, why? What is the motivation?

2)Does any relation bewteen the connection at here with the differential geometry(Riemannian geometry or fibre bundles)?

3)A detail: the conditions has a matrix form with graded commutator, I have no clear rule of it, how to do it?

We can find the connection at the book K-Theory for Operator Algebras by Bruce Blackadar,18.3 see here.

Answer 1

I believe the motivation for KK-theory comes from the Atiyah-Singer theorem - the idea is to describe certain differential operators on manifolds as Fredholm operators. There are perhaps three ways to understand KK-theory, depending on your inclination :

1. Kasparov's original definition - the idea behind it is in Higson/Roe's book "Analytic K-homology", and the details are in Jensen/Thomsen's book.
2. Cuntz' definition using quasi homomorphisms, which he discusses in a paper "Generalized homomorphisms between C* algebras and KK-theory"
3. Connes/Higson's E-theory, which they discuss in a paper "Almost homomorphisms and KK-theory". E-theory is a universal bi-functor on C* algebras which coincides with KK-theory for nuclear C* algebras.

As for connections, I have no clue. Sorry!