On positively curved 4-manifolds with S^1-symmetry

TL;DR

This paper proves that positively curved simply connected 4-manifolds with S^1-symmetry are diffeomorphic to S^4 or CP^2, extending known topological results to smooth classifications and revealing additional symmetry structures.

Contribution

It establishes a diffeomorphism classification for such 4-manifolds and shows they admit smooth torus actions equivalent to linear actions on standard models.

Findings

Positively curved simply connected 4-manifolds with circle symmetry are diffeomorphic to S^4 or CP^2.

Such manifolds admit extended torus actions that are smoothly equivalent to linear actions.

The classification uses the Pao replacement trick to analyze circle action configurations.

Abstract

It is well-known by the work of Hsiang and Kleiner that every closed oriented positively curved 4-dimensional manifold with an effective isometric S^1-action is homeomorphic to S^4 or CP^2. As stated, it is a topological classification. The primary goal of this paper is to show that it is indeed a diffeomorphism classification for such 4-dimensional manifolds. The proof of this diffeomorphism classification also shows an even stronger statement that every positively curved simply connected 4-manifold with an isometric circle action admits another smooth circle action which extends to a 2-dimensional torus action and is equivariantly diffeomorphic to a linear action on S^4 or CP^2. The main strategy is to analyze all possible topological configurations of effective circle actions on simply connected 4-manifolds by using the so-called replacement trick of Pao.