Mapping Analytic Surgery to Homology, Higher Rho Numbers and Metrics of Positive Scalar Curvature

Let Γ be a finitely generated discrete group and let M' be a Galois Γ-covering of a smooth compact manifold M. Let u: M → BΓ be the associated classifying map. Finally, let S^Γ_∗ (M') be the analytic structure group, a K-theory group appearing in the Higson-Roe analytic surgery sequene. Under suitable assumptions on the group Γ we construct two pairings, first between S^Γ_∗ (M') and the delocalized part of the cyclic cohomology of CΓ and secondly between S^Γ_∗ (M') and the relative cohomology H_{∗−1}(M → BΓ). Both 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(D) ∈ S^Γ_∗ (M') of an invertible Γ-equivariant Dirac type operator on M'. This applies, for example, to the rho class ρ(g) of a positive scalar curvature metric g on M, if M is spin. Regarding the first pairing, we establish in fact a more general result. Indeed, for an arbitrary discrete group Γ we prove that it is possible to map the whole Higson-Roe analytic surgery sequence to a long exact sequence in even/odd-graded non-commutative de Rham homology associated to a holomorphically closed subalgebra of C^∗ Γ, denoted AΓ . In particular, we prove that there exists a group homomorphism S^Γ_∗ (M') → H_{del} (AΓ). We then establish that under additional assumptions on Γ there exists a pairing between the delocalized part of the cyclic cohomology of CΓ and H_{del}(AΓ). We give a detailed study of the action of the diffeomorphism group on the analytic surgery sequence and on its homological counterpart, proving in particular precise formulae for the behaviour of the higher rho numbers under the action of the diffeomorphism group. We use these formulae in order to establish new lower bounds on the size of the moduli space of metrics of positive scalar curvature.