Journal of Noncommutative Geometry

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Volume 9, Issue 3, 2015, pp. 821–850
DOI: 10.4171/JNCG/209

Published online: 2015-10-29

Spectral sections, twisted rho invariants and positive scalar curvature

Moulay-Tahar Benameur[1] and Varghese Mathai[2]

(1) Université Montpellier 2, France
(2) University of Adelaide, Australia

We had previously defined in [10], the rho invariant $\rho_{spin}(Y,\Epsilon,H, g)$ for the twisted Dirac operator $\dirac^\Epsilon_H$ on a closed odd dimensional Riemannian spin manifold $(Y, g)$, acting on sections of a flat hermitian vector bundle $\Epsilon$ 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 $\rho_{spin}(Y,\Epsilon,H, g) - \rho_{spin}(Y,\Epsilon, g)$ in terms of spectral flows and prove that $\rho_{spin}(Y,\Epsilon,H, g) \in \mathbb Q$, 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 $\rho_{spin}(Y,\Epsilon,H, rg) =0$ for all $r \gg 0$, significantly generalizing results in [10]. These results are proved using the Bismut–Weitzenböck formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson–Roe approach [22].

Keywords: Twisted Dirac rho invariant, twisted Dirac eta invariant, conformal invariants, twisted Dirac operator, positive scalar curvature, manifolds with boundary, maximal Baum– Connes conjecture, vanishing theorems, spectral sections, spectral flow, structure groups, K-theory

Benameur Moulay-Tahar, Mathai Varghese: Spectral sections, twisted rho invariants and positive scalar curvature. J. Noncommut. Geom. 9 (2015), 821-850. doi: 10.4171/JNCG/209