Positive scalar curvature on simply connected spin pseudomanifolds

Let M-Sigma be an n-dimensional Thom-Mather stratified space of depth 1. We denote by beta M the singular locus and by L 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 alpha-class alpha(omega)(M-Sigma) is an element of KOn. In order to establish a sufficient condition, we need to assume additional structure: we assume that the link of M-Sigma is a homogeneous space of positive scalar curvature, L = G/K, where the semisimple compact Lie group G acts transitively on L by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when MS and beta M are spin, we reinterpret our obstruction in terms of two a-classes associated to the resolution of M-Sigma, M, and to the singular locus beta M. Finally, when M-Sigma, beta M, L and G are simply connected and dimM is big enough, and when some other conditions on L (satisfied in a large number of cases) hold, we establish the main result of this paper, showing that the vanishing of these two a-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.