Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature
Paolo Piazza, Thomas Schick, Vito Felice Zenobi
TL;DR
The paper develops a bridge between the Higson–Roe analytic surgery framework and noncommutative de Rham homology by mapping the analytic surgery sequence to a long exact sequence in $H_*(\mathcal A\Gamma)$ for a suitable dense subalgebra $\mathcal A\Gamma$. It defines higher rho numbers by pairing rho classes with delocalized cyclic cocycles and, under geometric and group-theoretic assumptions (e.g., hyperbolic groups or polynomial growth), constructs robust pairings with both delocalized and relative cohomology, including $H^*(M\to B\Gamma)$. The authors connect the delocalized Chern character of rho classes to Lott’s higher eta/invariant and prove a delocalized Atiyah–Patodi–Singer-type relation in this setting, providing concrete formulas for higher rho invariants and their behavior under diffeomorphisms. The work yields new tools to study the moduli space of positive scalar curvature metrics, showing, in several cases, that $\pi_0(\mathcal R^+(M)/\mathrm{Diff}(M))$ can be infinite, with ranks controlled by delocalized and relative group cohomology. Overall, this advances secondary index theory and its geometric applications by unifying operator algebra, K-theory, and noncommutative geometry methods with explicit delta-local invariants for broad classes of groups.
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.