The bordism group of unbounded KK-cycles

Abstract

We consider Hilsum’s notion of bordism as an equivalence relation on unbounded [Formula: see text]-cycles and study the equivalence classes. Upon fixing two [Formula: see text]-algebras, and a ∗-subalgebra dense in the first [Formula: see text]-algebra, a [Formula: see text]-graded abelian group is obtained; it maps to the Kasparov [Formula: see text]-group of the two [Formula: see text]-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 [Formula: see text]-algebra is the complex numbers (i.e. for [Formula: see text]-theory) and is a split surjection if the first [Formula: see text]-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense ∗-subalgebra.