Abstract:The $K$-homology groups of a $C^*$-algebra are receptacles for information from topology, operator algebra theory, and representation theory. For applications, one often wants to know if two $K$-homology classes are the same: the simplest way to deduce this is typically via the `primary' pairing between $K$-homology and the dual theory ($K$-theory). However, this pairing will typically miss some information: for example, it cannot detect torsion elements of $K$-homology.
In this paper, we introduce a `secondary' pairing between subgroups of $K$-homology and $K$-theory that takes values in $\mathbb{Q}/\mathbb{Z}$. In good cases we show that this pairing will detect all the classes in $K$-homology that are missed by the primary pairing. We then relate our secondary pairing to the relative eta invariants of Atiyah-Patodi-Singer, and to the Thomsen exact sequence and zeta maps from $C^*$-algebra classification theory.
| Subjects: | K-Theory and Homology (math.KT); Operator Algebras (math.OA) |
| MSC classes: | 19K33, 19K35, 46L35, 46L80, 46L85, 58J28 |
| Cite as: | arXiv:2605.23010 [math.KT] |
| (or arXiv:2605.23010v1 [math.KT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.23010 arXiv-issued DOI via DataCite (pending registration) |
From: Rufus Willett [view email]
[v1]
Thu, 21 May 2026 20:31:52 UTC (33 KB)
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