Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants

Abstract

In this paper, we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space 𝑀Σ with singular stratum 𝛽𝑀 (a closed manifold of positive codimension) and associated link equal to 𝐿, a smooth compact manifold. We briefly call such spaces manifolds with 𝐿-fibered singularities. Under suitable spin assumptions, we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that 𝐿 is a simply connected homogeneous space of positive scalar curvature, 𝐿 = 𝐺/𝐻, with the semisimple compact Lie group 𝐺 acting transitively on 𝐿 by isometries, we investigate when these necessary conditions are also sufficient. Our main result is that our conditions are indeed enough for large classes of examples, even when 𝑀Σ and 𝛽𝑀 are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.