Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotients

TL;DR

This paper investigates the topology of the space of positive Ricci and non-negative sectional curvature metrics on certain exotic spheres and sphere bundles, revealing infinitely many disconnected components in their moduli spaces.

Contribution

It classifies the moduli space components of curvature metrics on sphere bundles over spheres and quotients of exotic spheres, showing they have infinitely many path components.

Findings

Moduli space of positive Ricci metrics on certain sphere bundles has infinitely many components.

Quotients of Milnor spheres have infinitely many components in their non-negative sectional curvature metrics.

Finiteness results for quotients of Shimada spheres with infinitely many manifolds in the family.

Abstract

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of $S^7$-bundles over $S^8$ which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an infinite family of manifolds, the moduli space of positive Ricci curvature metrics has infinitely many path components.