Bordisms and unbounded $KK$-theory

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.