On the topology of the moduli space of positive scalar curvature concordances

Abstract

Let $M$ be a manifold which admits a metric with positive scalar curvature (or a positive intermediate curvature in a suitable sense). We study the moduli space ${\mathcal{M}}_{\bigsqcup }^{{\mathsf{pos}}_{\ast }}(M\times I{)}_g$ of concordances of such metrics (with appropriate boundary conditions) which restrict to a given metric $g$ on $M\times \{0\}\cup \partial M\times I$. We show that ${\pi }_{4\ast }{\mathcal{M}}_{\bigsqcup }^{{\mathsf{pos}}_{\ast }}(M\times I{)}_g\otimes \mathbb{Q}\ne 0$ in a stable range provided $\dim M$ is even. We obtain analogous results when positive scalar curvature is replaced by $k$-positive Ricci curvature for $k\ge 2$.

25 pages. Small exposition changes