Moduli spaces of nonnegatively curved metrics on exotic spheres

TL;DR

This paper demonstrates that the moduli space of nonnegatively curved metrics on certain 7-manifolds, including spheres, has infinitely many connected components distinguished by the Kreck-Stolz s-invariant, using advanced index theorem techniques.

Contribution

It introduces methods to compute the Kreck-Stolz s-invariant for metrics on orbifold bundles, revealing the rich structure of the moduli space on exotic spheres.

Findings

The moduli space has infinitely many connected components.

The s-invariant distinguishes different components.

New techniques for index calculations on orbifolds with group actions.

Abstract

We show that the moduli space of nonnegatively curved metrics on each member of a large class of 2-connected 7-manifolds, including each smooth manifold homeomorphic to $S^7$, has infinitely many connected components. The components are distinguished using the Kreck-Stolz $s$-invariant computed for metrics constructed by Goette, Kerin and Shankar. The invariant is computed by extending each metric to the total space of an orbifold disc bundle and applying generalizations of the Atiyah-Patodi-Singer index theorem for orbifolds with boundary. We develop methods for computing characteristic classes and integrals of characteristic forms appearing in index theorems for orbifolds, in particular orbifolds constructed using Lie group actions of cohomogeneity one.