Geometric non-commutative geometry

TL;DR

This paper extends non-existence results for positive scalar curvature metrics on foliated manifolds, including non-compact cases, using non-commutative geometric methods and cohomological obstructions.

Contribution

It generalizes previous non-existence theorems to non-compact manifolds with bounded geometry and introduces a new obstruction based on the A-hat class in Haefliger cohomology.

Findings

No positive scalar curvature metrics on certain compact foliations.

Extension of non-existence results to non-compact manifolds.

First obstruction for leafwise PSC using Haefliger cohomology.

Abstract

In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive scalar curvature. In this paper we review in more detail the material needed for the proof of our theorem and we extend our non-existence results to non-compact manifolds of bounded geometry. We also give a first obstruction result for the existence of metric with (not necessarily uniform) leafwise PSC in terms of the A-hat class in Haefliger cohomology.