Positive scalar curvature on simply connected spin pseudomanifolds

Abstract

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 ${\alpha }_w(M_{\Sigma })\in K{O}_n$. 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 $M_{\Sigma }$ and $\beta M$ are spin, we reinterpret our obstruction in terms of two $\alpha$-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 $\dim M$ 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 article, showing that the vanishing of these two $\alpha$-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.

28 pages. A few minor corrections from previous version. To appear in Journal of Topology and Analysis

References in corpus (3)

• Singular spaces, groupoids and metrics of positive scalar curvature
• Positive scalar curvature on manifolds with fibered singularities
• Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants

Cited by in corpus (4)

• Positive scalar curvature on manifolds with fibered singularities
• The Smith Fiber Sequence and Invertible Field Theories
• Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants
• Homotopy invariance of the space of metrics with positive scalar curvature on manifolds with singularities