Paolo Piazza
(Universitá di Roma “Sapienza”)
March 10, 2021
14:00
(CET)
Organized by Europe GNCG Seminar.

## Surgery sequences and higher invariants of Dirac operators

The analysis of the signature operator and of the spin-Dirac operator on a $$\Gamma$$-Galois covering of a smooth compact manifold allows to map the surgery sequence in differential topology and the Stolz surgery sequence for positive scalar curvature metrics to the Higson-Roe analytic surgery sequence. The latter organizes in a very efficient way primary and secondary K-theory invariants of these two differential operators, in fact of any Dirac-type operator. In this talk, I will explain how it is possible to further map the Higson-Roesequence to a sequence built out of the non-commutative de Rham homology groups of a dense holomorphically closed subalgebra of $$C^*_r \Gamma$$. At this stage $$\Gamma$$ is any finitely generated discrete group.I will then report on how this mapping,  together with suitable hypothesis on the group $$\Gamma$$, can be used in order to define and study higher numeric invariants (primary and secondary) associated to Dirac operators; in particular, I will talk about higher rho numbers of an invertible Dirac operator when$\Gamma$ is Gromov hyperbolic. An application, new results on the moduli spaces of positive scalar curvature metrics will be given. This is joint work with Thomas Schick and Vito Felice Zenobi.