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

Abstract

Let $Γ$ be a f.g. discrete group and let $\tilde M$ be a Galois $Γ$-covering of a smooth closed manifold $M$. Let $S_*^Γ(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_*^Γ(\tilde M)\to K_*(M)\to K_*(C_r^*Γ)\to$. We prove that for an arbitrary discrete group $Γ$ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology $\to H_{[*-1]}(\mathcal{A}Γ)\to H^{del}_{[*-1]}(\mathcal{A}Γ)\to H^{e}_{[*]}(\mathcal{A}Γ)\to$, with $\mathcal{A}Γ$ a dense homomorphically closed subalgebra of $C^*_rΓ$. Here, $ H_{*}^{del}(\mathcal{A}Γ)$ is the delocalized homology and $H_{*}^{e}(\mathcal{A}Γ)$ is the homology localized at the identity element. Then, under additional assumptions on $Γ$, we prove the existence of a pairing between $HC^*_{del}(\mathbb{C}Γ)$, the delocalized part of the cyclic cohomology of $\mathbb{C}Γ$, and $H^{del}_{*-1}(\mathcal{A}Γ)$. This, in particular, gives a pairing between $S^Γ_*(\tilde M)$ and $HC^{*-1}_{del}(\mathbb{C}Γ)$. We also prove the existence of a pairing between $S^Γ_*(\tilde M)$ and the relative cohomology $H^{[*-1]}(M\to BΓ)$. 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 $ρ(\tilde D)\in S_*^Γ(\tilde M)$ of an invertible $Γ$-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.