On moduli spaces of positive scalar curvature metrics on highly connected manifolds

TL;DR

This paper investigates the topology of moduli spaces of positive scalar curvature metrics on highly connected, simply connected spin manifolds, revealing non-trivial higher homotopy groups and fundamental groups in specific cases.

Contribution

It demonstrates the non-triviality of higher homotopy groups and the fundamental group of the moduli space of positive scalar curvature metrics on certain highly connected manifolds.

Findings

Observer moduli space has non-trivial higher homotopy groups.

Fundamental group of the moduli space is non-trivial for certain manifolds.

Results apply to (2n-2)-connected (4n-2)-dimensional manifolds.

Abstract

Let $M$ be a simply connected spin manifold of dimension at least six which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar cuvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds which includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.