Geometric non-commutative geometry

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