12 nov 2014 -- 14:30
Aula di Consiglio, Dip. Matematica, Università "La Sapienza", Roma
Abstract.
For a discrete group $\Gamma$, the analytic surgery exact sequence of Higson and Roe characterizes the failure of the Baum-Connes assembly map $\mu: K_*(B\Gamma)\rightarrow K_*(C^*\Gamma)$ to be an isomorphism. In particular, there are obstruction structure groups $S_*(\Gamma)$ which vanish precisely when the assembly map $\mu$ is bijective. Higson and Roe showed that the certain spectral invariants associated with finite-dimensional representations of $\Gamma$ called relative eta-invariants (aka rho-invariants), defined in the seminal work of Atiyah, Patodi and Singer, appear naturally as morphisms from these structure groups to the reals. This allowed them to conceptualize and re-prove classical deep results of Mathai, Weinberger, Keswani, and Piazza-Schick on the vanishing of the Atiyah-Patodi-Singer rho-invariants for metrics of positive scalar curvature for spin Dirac operators and their homotopy invariance for signature operators, when $\Gamma$ is torsion-free and the assembly map $\mu$ is an isomorphism. We shall extend the utility of the analytic surgery exact sequence to the semi-finite case associated with Galois $\Gamma$-coverings, by introducing $\ell^2$-structure groups. These structure groups appear in an exact sequence which is compatible with the one studied by Higson and Roe. We thus prove the corresponding vanishing results for $\ell^2$ rho-invariants, first introduced and studied by Cheeger and Gromov. (joint work with M.-T. Benameur)