2019/05/28 by Piazza, Paolo, Schick, Thomas, Zenobi, Vito Felice · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1905.11861
Let Γ be a f.g. discrete group and let M be a Galois Γ-covering of a smooth closed manifold M. Let S_*Γ(M) be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence → S_*Γ( M)→ K_*(M)→ K_*(Cr^*Γ)→. 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 → H[*-1](AΓ)→ Hdel[*-1](AΓ)→ He[*](AΓ)→, with AΓ a dense homomorphically closed subalgebra of C^*rΓ. Here, H*del(AΓ) is the delocalized homology and H*e(AΓ) is the homology localized at the identity element. Then, under additional assumptions on Γ, we prove the existence of a pairing between HC^*del(ℂΓ), the delocalized part of the cyclic cohomology of ℂΓ, and Hdel*-1(AΓ). This, in particular, gives a pairing between SΓ_*( M) and HC*-1del(ℂΓ). We also prove the existence of a pairing between SΓ_*( M) and the relative cohomology H[*-1](M→ 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( D)∈ S_*Γ( M) of an invertible Γ-equivariant Dirac type operator on 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.