Nonconnected Moduli Spaces of Nonnegative Sectional Curvature Metrics on Simply Connected Manifolds

TL;DR

This paper demonstrates that in dimensions of the form 4n+3, there are infinitely many simply connected manifolds with nonnegative sectional curvature whose moduli spaces of such metrics are highly disconnected, extending previous results beyond dimension seven.

Contribution

It establishes the existence of infinitely many simply connected manifolds in each dimension 4n+3 with nonnegative sectional curvature moduli spaces having infinitely many components, generalizing earlier findings.

Findings

Infinite sequences of manifolds with disconnected moduli spaces in each dimension 4n+3.

Extension of disconnected moduli space results from dimension seven to higher dimensions.

Application to moduli spaces of positive Ricci curvature and open manifolds.

Abstract

We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result does also hold for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such $M$ the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold $M\times\mathbb {R}$ also has infinitely many components.