Spectral sections, twisted rho invariants and positive scalar curvature

Moulay Tahar Benameur, Varghese Mathai
2015 Journal of Noncommutative Geometry  
We had previously defined the rho invariant ρ_spin(Y,E,H, g) for the twisted Dirac operator ∂^E_H on a closed odd dimensional Riemannian spin manifold (Y, g), acting on sections of a flat hermitian vector bundle E over Y, where H = ∑ i^j+1 H_2j+1 is an odd-degree differential form on Y and H_2j+1 is a real-valued differential form of degree 2j+1. Here we show that it is a conformal invariant of the pair (H, g). In this paper we express the defect integer ρ_spin(Y,E,H, g) - ρ_spin(Y,E, g) in
more » ... spin(Y,E, g) in terms of spectral flows and prove that ρ_spin(Y,E,H, g)∈ Q, whenever g is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for π_1(Y) (which is assumed to be torsion-free), then we show that ρ_spin(Y,E,H, rg) =0 for all r≫ 0, significantly generalizing our earlier results. These results are proved using the Bismut-Weitzenböck formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson-Roe approach.
doi:10.4171/jncg/209 fatcat:zjvsu2c7qfdc7kk7uzkr63tltm