Spectral sections, twisted rho invariants and positive scalar curvature

Abstract

We had previously defined the rho invariant ${\rho }_{spin}(Y,E,H,g)$ for the twisted Dirac operator $/{\partial }_H^E$ on a closed odd dimensional Riemannian spin manifold $(Y,g)$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H=\sum i^{j+1}{H}_{2j+1}$ 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 ${\rho }_{spin}(Y,E,H,g)-{\rho }_{spin}(Y,E,g)$ in terms of spectral flows and prove that ${\rho }_{spin}(Y,E,H,g)\in \mathbb{Q}$, whenever $g$ is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for ${\pi }_1(Y)$ (which is assumed to be torsion-free), then we show that ${\rho }_{spin}(Y,E,H,rg)=0$ for all $r\gg 0$, 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.

25 pages. Minor corrections made, but no changes to the results

References in corpus (6)

• Index, eta and rho-invariants on foliated bundles
• Analytic Torsion of Z_2-graded Elliptic Complexes
• On the noncommutative spectral flow
• Twisted Analytic Torsion
• Conformal invariants of twisted Dirac operators and positive scalar curvature
• Index type invariants for twisted signature complexes and homotopy invariance

Cited by in corpus (6)

• Singular spaces, groupoids and metrics of positive scalar curvature
• Conformal invariants of twisted Dirac operators and positive scalar curvature
• Index type invariants for twisted signature complexes and homotopy invariance
• The Higson-Roe exact sequence and ${\ell }^2$ eta invariants
• Positive scalar curvature via end-periodic manifolds
• An equivariant PPV theorem and Paschke-Higson duality