Spectral sections, twisted rho invariants and positive scalar curvature
Date
2015
Authors
Benameur, M.
Mathai, V.
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Journal article
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Journal of Noncommutative Geometry, 2015; 9(3):821-850
Statement of Responsibility
Moulay Tahar Benameur, and Varghese Mathai
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Abstract
<jats:p>
We had previously defined in [10], the rho invariant
<jats:inline-formula>
<jats:tex-math>\rho_{spin}(Y,\Epsilon,H, g)</jats:tex-math>
</jats:inline-formula>
for the twisted Dirac operator
<jats:inline-formula>
<jats:tex-math>\def\partialslash{\mathrlap{/}{\partial}}\partialslash^{\mathcal E}_H</jats:tex-math>
</jats:inline-formula>
on a closed odd dimensional Riemannian spin manifold
<jats:inline-formula>
<jats:tex-math>(Y, g)</jats:tex-math>
</jats:inline-formula>
, acting on sections of a flat hermitian vector bundle
<jats:inline-formula>
<jats:tex-math>\Epsilon</jats:tex-math>
</jats:inline-formula>
over
<jats:inline-formula>
<jats:tex-math>Y</jats:tex-math>
</jats:inline-formula>
, where
<jats:inline-formula>
<jats:tex-math>H = \sum i^{j+1} H_{2j+1}</jats:tex-math>
</jats:inline-formula>
is an odd-degree differential form on
<jats:inline-formula>
<jats:tex-math>Y</jats:tex-math>
</jats:inline-formula>
and
<jats:inline-formula>
<jats:tex-math>H_{2j+1}</jats:tex-math>
</jats:inline-formula>
is a real-valued differential form of degree
<jats:inline-formula>
<jats:tex-math>{2j+1}</jats:tex-math>
</jats:inline-formula>
. Here we show that it is a conformal invariant of the pair
<jats:inline-formula>
<jats:tex-math>(H, g)</jats:tex-math>
</jats:inline-formula>
. In this paper we express the defect integer
<jats:inline-formula>
<jats:tex-math>\rho_{spin}(Y,\Epsilon,H, g) - \rho_{spin}(Y,\Epsilon, g)</jats:tex-math>
</jats:inline-formula>
in terms of spectral flows and prove that
<jats:inline-formula>
<jats:tex-math>\rho_{spin}(Y,\Epsilon,H, g) \in \mathbb Q</jats:tex-math>
</jats:inline-formula>
, whenever
<jats:inline-formula>
<jats:tex-math>g</jats:tex-math>
</jats:inline-formula>
is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum–Connes conjecture holds for
<jats:inline-formula>
<jats:tex-math>\pi_1(Y)</jats:tex-math>
</jats:inline-formula>
(which is assumed to be torsion-free), then we show that
<jats:inline-formula>
<jats:tex-math>\rho_{spin}(Y,\Epsilon,H, rg) =0</jats:tex-math>
</jats:inline-formula>
for all
<jats:inline-formula>
<jats:tex-math>r \gg 0</jats:tex-math>
</jats:inline-formula>
, 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].
</jats:p>
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