A secondary pairing between K-theory and K-homology, relative eta invariants, and zeta maps
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| Format: | Preprint |
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2026
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| _version_ | 1866914590454972416 |
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| author | Willett, Rufus |
| author_facet | Willett, Rufus |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23010 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A secondary pairing between K-theory and K-homology, relative eta invariants, and zeta maps Willett, Rufus K-Theory and Homology Operator Algebras 19K33, 19K35, 46L35, 46L80, 46L85, 58J28 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. |
| title | A secondary pairing between K-theory and K-homology, relative eta invariants, and zeta maps |
| topic | K-Theory and Homology Operator Algebras 19K33, 19K35, 46L35, 46L80, 46L85, 58J28 |
| url | https://arxiv.org/abs/2605.23010 |