$\mathsf S^1$-actions on $4$-manifolds and fixed point homogeneous manifolds of nonnegative curvature

TL;DR

This paper investigates the structure of 4-manifolds with nonnegative curvature under circle actions, showing their quotients can be approximated by positively curved 3-manifolds and classifying fixed point homogeneous manifolds.

Contribution

It demonstrates that quotients of certain 4-manifolds can be approximated by smooth positively curved 3-manifolds and classifies fixed point homogeneous manifolds in this setting.

Findings

Quotients admit Gromov-Hausdorff approximations by smooth positively curved 3-manifolds.

Fixed point homogeneous manifolds are diffeomorphic to normal bundles of submanifolds.

Torus manifolds with nonnegative curvature are rationally elliptic.

Abstract

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively curved Riemannian 3-manifolds. In the second main part we show that a compact, nonnegatively curved and fixed point homogeneous manifold is diffeomorphic to the unit normal bundles of two smooth closed submanifolds glued together along their boundaries. As a corollary we show that a compact and simply connected torus manifold admitting an invariant metric of nonnegative curvature is rationally elliptic.