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

TL;DR

This paper investigates the topology of the space of concordances of positive scalar curvature metrics on manifolds, revealing nontrivial rational homotopy groups in certain dimensions and extending results to k-positive Ricci curvature.

Contribution

It demonstrates nontrivial rational homotopy groups of the concordance space for positive scalar curvature metrics, including cases with k-positive Ricci curvature, in even-dimensional manifolds.

Findings

Nontrivial rational homotopy groups in stable ranges for even-dimensional manifolds.

Extension of results from scalar curvature to k-positive Ricci curvature.

Identification of non-vanishing homotopy groups in the moduli space of 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 ${\mathscr{M}}^{{\mathsf{pos}}_*}_{\sqcup}(M\times I)_g$ 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 $\pi_{4*}{\mathscr{M}}^{{\mathsf{pos}}_*}_{\sqcup}(M \times I)_g \otimes {\mathbb Q} \neq 0$ 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 $k \ge 2$.