A secondary pairing between K-theory and K-homology, relative eta invariants, and zeta maps

The $K$-homology groups of a $C^{\ast }$-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 $\mathbb{Q}\mathrm{/}\mathbb{Z}$. 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 $C^{\ast }$-algebra classification theory.