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

Abstract

Let $\Gamma$ be a f.g. discrete group and let $\tilde M$ be a Galois $\Gamma$-covering of a smooth closed manifold $M$. Let $S_*^\Gamma(\tilde{M})$ 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 $\Gamma$ 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 $\mathcal{A}\Gamma$ a dense homomorphically closed subalgebra of $C^*_r\Gamma$. Here, $ H_{*}^{del}(\mathcal{A}\Gamma)$ is the delocalized homology and $H_{*}^{e}(\mathcal{A}\Gamma)$ is the homology localized at the identity element. Then, under additional assumptions on $\Gamma$, we prove the existence of a pairing between $HC^*_{del}(\mathbb{C}\Gamma)$, the delocalized part of the cyclic cohomology of $\mathbb{C}\Gamma$, and $H^{del}_{*-1}(\mathcal{A}\Gamma)$. This, in particular, gives a pairing between $S^\Gamma_*(\tilde M)$ and $HC^{*-1}_{del}(\mathbb{C}\Gamma)$. We also prove the existence of a pairing between $S^\Gamma_*(\tilde M)$ and the relative cohomology $H^{[*-1]}(M\to B\Gamma)$. 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 $\rho(\tilde D)\in S_*^\Gamma(\tilde M)$ of an invertible $\Gamma$-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.