Positive scalar curvature on manifolds with fibered singularities

Abstract

A (compact) manifold with fibered $P$-singularities is a (possibly) singular pseudomanifold $M_\Sigma$ with two strata: an open nonsingular stratum $\mathring M$ (a smooth open manifold) and a closed stratum $\beta M$ (a closed manifold of positive codimension), such that a tubular neighborhood of $\beta M$ is a fiber bundle with fibers each looking like the cone on a fixed closed manifold $P$. We discuss what it means for such an $M_{\Sigma}$ with fibered $P$-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type $P$ is either $\mathbb Z/k$ or $S^1$, and $M$ and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin$^c$ manifolds with positive "twisted scalar curvature," where the twisting comes from the curvature of the spin$^c$ line bundle.