The bordism group of unbounded KK-cycles

Abstract

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

arXiv comment38 pages