Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature

Let $\mathrm{\Gamma }$ be a f.g. discrete group and let $\tilde{M}$ be a Galois $\mathrm{\Gamma }$-covering of a smooth closed manifold $M$. Let $S_{\ast }^{\mathrm{\Gamma }}(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_^\Gamma(\tilde M)\to K_(M)\to K_(C_r^\Gamma)\to$. We prove that for an arbitrary discrete group $\mathrm{\Gamma }$ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology $\to H_{[-1]}($\mathcal{A}$\Gamma)\to H^{del}_{[-1]}($\mathcal{A}$\Gamma)\to H^{e}_{[]}($\mathcal{A}$\Gamma)\to$,with$$\mathcal{A}$\Gamma$adensehomomorphicallyclosedsubalgebraof$C^_r\Gamma$. Here, $H_{\ast }^{del}(\mathcal{A}\mathrm{\Gamma })$ is the delocalized homology and $H_{\ast }^e(\mathcal{A}\mathrm{\Gamma })$ is the homology localized at the identity element. Then, under additional assumptions on $\mathrm{\Gamma }$, we prove the existence of a pairing between $H{C}_{del}^{\ast }(\mathbb{C}\mathrm{\Gamma })$, the delocalized part of the cyclic cohomology of $\mathbb{C}\mathrm{\Gamma }$, and $H_{\ast -1}^{del}(\mathcal{A}\mathrm{\Gamma })$. This, in particular, gives a pairing between $S_{\ast }^{\mathrm{\Gamma }}(\tilde{M})$ and $H{C}_{del}^{\ast -1}(\mathbb{C}\mathrm{\Gamma })$. We also prove the existence of a pairing between $S_{\ast }^{\mathrm{\Gamma }}(\tilde{M})$ and the relative cohomology $H^{[\ast -1]}(M\to B\mathrm{\Gamma })$. Both these parings are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class $\rho (\tilde{D})\in S_{\ast }^{\mathrm{\Gamma }}(\tilde{M})$ of an invertible $\mathrm{\Gamma }$-equivariant Dirac type operator on $\tilde{M}$. Finally, we provide a precise study for the behavior of all previous K-theoretic and homological objects and of the higher rho numbers under the action of the diffeomorphism group of $M$. Then, we establish new results on the moduli space of metrics of positive scalar curvature when $M$ is spin.