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arXiv · 1503.07398

The bordism group of unbounded KK-cycles

Robin J. Deeley·Magnus Goffeng·Bram Mesland
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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.

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Robin J. Deeley, Magnus Goffeng, Bram Mesland. 2015-03-25. The bordism group of unbounded KK-cycles. https://arxiv.org/abs/1503.07398

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