Mathematics > K-Theory and Homology
[Submitted on 25 Mar 2015 (v1), last revised 27 Jul 2016 (this version, v2)]
Title:The bordism group of unbounded KK-cycles
View PDFAbstract: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.
Submission history
From: Magnus Goffeng [view email][v1] Wed, 25 Mar 2015 14:43:18 UTC (37 KB)
[v2] Wed, 27 Jul 2016 08:55:10 UTC (39 KB)
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