H-Spaces, Loop Spaces and the Space of Positive Scalar Curvature Metrics on the Sphere

TL;DR

This paper demonstrates that for spheres of dimension n ≥ 3, the space of positive scalar curvature metrics has a rich algebraic structure, being homotopy equivalent to an n-fold loop space in certain cases.

Contribution

It establishes the homotopy equivalence of the space of positive scalar curvature metrics on spheres to an n-fold loop space, revealing new algebraic and topological structures.

Findings

The space is homotopy equivalent to an H-space with a connected sum based product.

An action of the little n-disks operad is constructed on this space.

For n=3 or n≥5, the space is weakly homotopy equivalent to an n-fold loop space.

Abstract

For dimensions n greater than or equal to 3, we show that the space of metrics of positive scalar curvature on the n-sphere is homotopy equivalent to a subspace which takes the form of a H-space with a homotopy commutative, homotopy associative product operation. This product operation is based on the connected sum construction. We then exhibit an action of the little n-disks operad on this subspace which, using results of Boardman, Vogt and May implies that when n=3 or n is at least 5, the space of metrics of positive scalar curvature on the n-sphere is weakly homotopy equivalent to an n-fold loop space.