Abstract

Co-author: Paul Baum (Penn State University)

K-homology, the homology theory dual to K-theory, can be described in a number of quite distinct models. One of them is analytic, uses Kasparov's KK-theory, and is the home of index problems. Another one uses geometric cycles, going back to Baum and Douglas. A large part of index theory is concerned with the isomorphism between the geoemtric and the analytic model, and with Chern character transformations to (co)homology.

In applications to string theory, and for certain index problems, twisted versions of K-theory and K-homology play an essential role.
 
We will descirbe the general context, and then focus on two new models for twisted K-homology and their applications and relations. These aere again based on geometric cycles in the spirit of Baum and Douglas. We will include in particular precise discussions of the different ways to define and work with twists (for us, classified by elements of the third integral cohomology group of the base space in question).