Bordisms and unbounded $KK$-theory

TL;DR

This monograph develops a geometric framework for unbounded KK-theory using KK-bordism, enabling the application of manifold techniques to noncommutative geometries like orbifolds and foliations.

Contribution

It introduces a KK-bordism relation on unbounded KK-cycles that recovers Kasparov's KK-theory and is suitable for studying secondary invariants in noncommutative geometry.

Findings

Defines KK-bordism group for unbounded KK-cycles.

Provides a geometric approach to unbounded KK-theory.

Facilitates the study of secondary invariants in noncommutative spaces.

Abstract

This monograph studies $KK$-theory in its unbounded model. The central object is the $KK$-bordism group obtained by imposing the $KK$-bordism relation on unbounded $KK$-cycles. In the paradigm of noncommutative geometry, an unbounded $KK$-cycle is a noncommutative geometry in its own right and our approach allow for the study of mildly noncommutative geometries (orbifolds, foliations et cetera) as if they were closed manifolds. The techniques we introduce enable us to directly import manifold techniques and arguments into the important yet technical field of unbounded $KK$-theory. Recent decades has seen a tremendous progress in the study of the unbounded model for $KK$ as well as secondary invariants, the first motivated by refining computational tools in Kasparov's $KK$-theory and the second by applications to geometry and topology. The aim of this work is to provide a common framework for these two areas: equipping unbounded $KK$-cycles with a geometrically motivated relation recovering Kasparov's $KK$-theory that is computationally tractable for working with secondary invariants.