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

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

2008-10-07

arxiv.org/abs/0810.1232v1

Mathematics

Differential Geometry

updated version of the manuscript written in 2007

Scientific paper

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.

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