Duality pairings with the analytic structure group

Abstract

We construct a slant product ${\mathrm{S}}_p^{G\times H}(X\times Y)\otimes {\mathrm{K}}_{-q}({\bar{\mathfrak{c}}}^{\mathrm{red}}Y⋊H)\to {\mathrm{K}}_{p-q}({\mathrm{C}}_G^{\ast }X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson compactification taking values in the (equivariant) Roe algebra. This complements the slant products constructed in earlier work of Engel and the authors ( arXiv:1909.03777 [math.KT] ). The distinguishing feature of our new slant product is that it specializes to a duality pairing ${\mathrm{S}}_p^H(Y)\otimes {\mathrm{K}}_{-p}({\bar{\mathfrak{c}}}^{\mathrm{red}}(Y)⋊H)\to \mathbb{Z}$ which can be used to extract numerical invariants out of elements in the analytic structure group such as rho-invariants associated to positive scalar curvature metrics.

33 pages