Diffeomorphisms and positive curvature

TL;DR

This paper demonstrates the existence of infinite order elements in the homotopy groups of spaces of metrics with positive Ricci and sectional curvature on certain high-dimensional Spin manifolds with specific topological properties.

Contribution

It establishes the presence of infinite homotopy elements in the spaces of positive curvature metrics for a class of high-dimensional Spin manifolds with non-vanishing rational Pontryagin classes.

Findings

Existence of infinite order elements in homotopy groups of positive curvature metric spaces.

Applicable to high-dimensional Spin manifolds with specific topological conditions.

Advances understanding of the topology of spaces of positive curvature metrics.

Abstract

We prove the existence of elements of infinite order in the homotopy groups of the spaces $\mathcal{R}_{Ric>0}(M)$ and $\mathcal{R}_{sec>0}(M)$ of positive Ricci and positive sectional curvature, provided that $M$ is high-dimensional and Spin, admits such a metric and has a non-vanishing rational Pontryagin class.