On the topology of the space of Ricci-positive metrics

TL;DR

This paper investigates the topological structure of the space of Ricci-positive metrics on certain high-dimensional manifolds, revealing nontrivial rational homology under specific conditions related to the manifold's genus and dimension.

Contribution

It demonstrates the existence of nontrivial rational homology in the space of Ricci-positive metrics on connected sums of spheres, extending to manifolds with additional spin structures.

Findings

Nontrivial rational homology in the space of Ricci-positive metrics for large genus and specific dimensions.

Applicability of the results to manifolds with spin structures and Ricci-positive metrics.

Conditions on dimension and genus for the topological complexity of the metric space.

Abstract

We show that the space $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n})$ of metrics with positive Ricci curvature on the manifold $W^{2n}_g := \sharp^g (S^n \times S^n)$ has nontrivial rational homology if $n \not \equiv 3 \pmod 4$ and $g$ are both sufficiently large. The same argument applies to $\mathcal{R}^{\mathrm{pRc}}(W_g^{2n} \sharp N)$ provided that $N$ is spin and $W_g^{2n} \sharp N$ admits a Ricci positive metric.