# Moduli space of metrics of nonnegative sectional or positive Ricci curvature on homotopy real projective spaces

## TL;DR

This paper demonstrates that the moduli spaces of metrics with nonnegative sectional or positive Ricci curvature on certain homotopy real projective spaces have infinitely many path components, revealing complex geometric structures.

## Contribution

It establishes the existence of infinitely many path components in the moduli spaces of such metrics on specific homotopy real projective spaces, extending previous results to new dimensions.

## Key findings

- Moduli space of nonnegative sectional curvature on homotopy RP^5 has infinitely many components.
- In dimensions 4k+1, there are many homotopy RP^{4k+1}s with infinitely many components in positive Ricci curvature metrics.
- Examples previously known only in higher dimensions now shown in lower dimensions.

## Abstract

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy ${\mathbb {R}} P^5$ has infinitely many path components. We also show that in each dimension $4k+1$ there are at least $2^{2k}$ homotopy ${\mathbb {R}} P^{4k+1}$s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions $4k+3\geq 7$.