Positive scalar curvature on simply connected spin pseudomanifolds

Abstract

Let [Formula: see text] be an [Formula: see text]-dimensional Thom–Mather stratified space of depth [Formula: see text]. We denote by [Formula: see text] the singular locus and by [Formula: see text] the associated link. In this paper, we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge [Formula: see text]-class [Formula: see text]. In order to establish a sufficient condition, we need to assume additional structure: we assume that the link of [Formula: see text] is a homogeneous space of positive scalar curvature, [Formula: see text], where the semisimple compact Lie group [Formula: see text] acts transitively on [Formula: see text] by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when [Formula: see text] and [Formula: see text] are spin, we reinterpret our obstruction in terms of two [Formula: see text]-classes associated to the resolution of [Formula: see text], [Formula: see text], and to the singular locus [Formula: see text]. Finally, when [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are simply connected and [Formula: see text] is big enough, and when some other conditions on [Formula: see text] (satisfied in a large number of cases) hold, we establish the main result of this paper, showing that the vanishing of these two [Formula: see text]-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.