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.

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