Homotopy groups of the moduli space of metrics of positive scalar curvature

TL;DR

This paper demonstrates that the homotopy groups of the moduli space of positive scalar curvature metrics on spheres and other manifolds are non-trivial in many degrees, using advanced surgery techniques and exotic sphere families.

Contribution

It develops a family version of Gromov-Lawson surgery and applies it to exotic sphere families to reveal non-trivial homotopy groups of the moduli space.

Findings

Homotopy groups of the moduli space are non-trivial in many degrees.

Construction of manifolds with non-trivial higher homotopy groups of the quotient space.

Extension of surgery techniques to families of manifolds.

Abstract

We prove that for many degrees in a stable range the homotopy groups of the moduli space of metrics of positive scalar curvature on S^n and on other manifolds are non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to an exotic smooth families of spheres due to Hatcher. As described, this works for all manifolds of suitable dimension and for the quotient of the space of metrics of positive scalar curvature by the (free) action of the subgroup of diffeomorphisms which fix a point and its tangent space. We also construct special manifolds where the quotient of the space of metrcis of positive scalar curvature by the full diffeomorphism group has non-trivial higher homotopy groups.