Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 062, 39 pages      arXiv:2005.02744      https://doi.org/10.3842/SIGMA.2021.062
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

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

Boris Botvinnik a, Paolo Piazza b and Jonathan Rosenberg c
a) Department of Mathematics, University of Oregon, Eugene OR 97403-1222, USA
b) Dipartimento di Matematica ''Guido Castelnuovo'', Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
c) Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

Received May 26, 2020, in final form June 08, 2021; Published online June 24, 2021

Abstract
In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_\Sigma$ with singular stratum $\beta M$ (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 sufficient for large classes of examples, even when $M_\Sigma$ and $\beta M$ are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.

Key words: positive scalar curvature; pseudomanifold; singularity; bordism; transfer; $K$-theory; index; rho-invariant.

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