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Classification of circle actions on 4-manifolds

This article studies locally smooth S l-actions on closed oriented 4-manifolds in terms of the orbit space, orbit type data, and the characteristic class of the action which lies inDHI(M*, S*) where M* is the orbit space and S* is the image of a certain collection of singular orbits. It is proved that such actions are determined by their weighted orbit spaces and are in 1-1 correspondence with "legally-weighted" 3-manifolds. The information contained in the weighted orbit space is used to give a presentation of the fundamental group of the 4-manifold, and in certain cases the quadratic form is computed. The classification of actions of the circle on 3-manifolds due to Seifert, Orlik, and Raymond [12], [11], [7] has been known for some years and has been widely applied in the study of other transformation groups on 3-manifolds. More recently, several papers have brought into focus certain aspects of actions of the circle on 4-manifolds. In particular there are the papers of Hsiang [5] and Church and Lamotke [2] which treat various cases of semifree circle actions on orientable 4-manifolds. Orlik and Wagreich [9] have studied the problem from the point of view of algebraic geometry, and Pao [10] has proved the existence of nonlinear circle actions on the 4-sphere. This article is a continuation of [4] in which we studied locally smooth actions of the circle group S' on simply connected 4-manifolds. In that paper an equivariant classification of these actions was given, and it was shown how to compute the quadratic forms of these 4-manifolds in terms of the classifying data. There resulted the theorem that any simply connected closed 4-manifold which carries a locally smooth S '-action is homotopy equivalent to a connected sum of the manifolds S4, CP2, Cp2, and S2 X S2. We improve this in ? 13. The present article considers classification of locally smooth effective S'-actions on closed oriented 4-manifolds in terms of the weighted orbit space, which consists of the orbit space, orbit type data, and the characteristic class of the action, which relates the Euler class of the principal orbit bundle to certain equivariant gluing information. Oriented S '-manifolds are called Received by the editors October 27, 1976. AMS (MOS) subject classifications (1970). Primary 57E25.