H-Space and Loop Space Structures for Intermediate Curvatures

TL;DR

This paper demonstrates that the space of metrics with positive Ricci curvature on spheres has an $H$-space structure and is homotopy equivalent to an $n$-fold loop space, revealing deep topological properties of curvature-constrained metrics.

Contribution

It establishes an $H$-space structure on the space of $k$-positive Ricci curvature metrics and links it to loop space theory using operads and homotopy theory.

Findings

The space of $k$-positive Ricci curvature metrics on spheres forms an $H$-space.

The path component containing the round metric is weakly homotopy equivalent to an $n$-fold loop space.

The structure is shown using operad theory and homotopy-theoretic methods.

Abstract

For dimensions $n\geq 3$ and $k\in\{2, \cdots, n\}$, we show that the space of metrics of $k$-positive Ricci curvature on the sphere $S^{n}$ has the structure of an $H$-space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May that the path component of this space containing the round metric is weakly homotopy equivalent to an $n$-fold loop space.