# Positive scalar curvature on simply connected spin pseudomanifolds

## TL;DR

This paper investigates conditions under which simply connected spin pseudomanifolds with stratified structure admit positive scalar curvature wedge metrics, linking index theory obstructions to geometric and topological properties of the space.

## Contribution

It establishes new obstructions and sufficient conditions for positive scalar curvature on stratified pseudomanifolds, connecting index theory with geometric structures of links and singular loci.

## Key findings

- Obstruction to positive scalar curvature expressed via wedge α-class in KO-theory.
- Sufficient conditions for existence of positive scalar curvature metrics involving vanishing of α-classes.
- Main theorem: vanishing of two α-classes implies existence of positive scalar curvature wedge metric.

## 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 $\alpha_w (M_\Sigma)\in KO_n$. 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.