The bordism group of unbounded KK-cycles

TL;DR

This paper investigates the structure of the bordism group of unbounded KK-cycles, establishing its relationship with Kasparov KK-theory and analyzing properties of the associated maps in various contexts.

Contribution

It introduces a new perspective on bordism classes of unbounded KK-cycles and studies the properties of the resulting abelian group and its relation to KK-theory.

Findings

The constructed group maps isomorphically to KK-theory when the first algebra is complex numbers.

It is a split surjection for continuous functions on compact manifolds with boundary.

Properties of the map are analyzed in both general and specific cases.

Abstract

We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a $\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first $C^*$-algebra is the complex numbers (i.e., for $K$-theory) and is a split surjection if the first $C^*$-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense $*$-subalgebra.