ON POSITIVELY CURVED FOUR-MANIFOLDS WITH S1-SYMMETRY

It is well known by the work of Hsiang and Kleiner that every closed oriented positively curved four-dimensional manifold with an effective isometric S1-action is homeomorphic to S4 or CP2. 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 four-dimensional manifolds. The proof of this diffeomorphism classification also shows an even stronger statement that every positively curved simply connected four-manifold with an isometric circle action admits another smooth circle action which extends to a two-dimensional torus action and is equivariantly diffeomorphic to a linear action on S4 or CP2. The main strategy is to analyze all possible topological configurations of effective circle actions on simply connected four-manifolds by using the so-called replacement trick of Pao.