Enlargeability, foliations, and positive scalar curvature

TL;DR

This paper extends fundamental results connecting geometry, characteristic numbers, and positive scalar curvature to spin foliations and enlargeable manifolds, providing new obstructions and vanishing theorems.

Contribution

It generalizes classical theorems to foliations, establishing new obstructions to positive scalar curvature and extending Connes' vanishing theorem.

Findings

Spin foliations on enlargeable manifolds admit no PSC metrics.

Any PSC metric on such foliations is bounded by the reciprocal of the foliation K-area.

Connes' vanishing theorem extends to Haefliger cohomology classes.

Abstract

We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold admits no PSC metric; that any metric of PSC on such a foliation is bounded by a multiple of the reciprocal of the foliation K-area of the ambient manifold; and that Connes' vanishing theorem for characteristic numbers of PSC foliations extends to a vanishing theorem for Haefliger cohomology classes.