Classification of circle actions on 4-manifolds

Fintushel, Ronald

doi:10.1090/S0002-9947-1978-0496815-7

Classification of circle actions on $4$-manifolds
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Trans. Amer. Math. Soc. 242 (1978), 377-390 Request permission

Abstract:

This article studies locally smooth ${S^1}$-actions on closed oriented 4-manifolds in terms of the orbit space, orbit type data, and the characteristic class of the action which lies in ${H_1}({M^{\ast }},{S^{\ast }})$ where ${M^{\ast }}$ is the orbit space and ${S^{\ast }}$ 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.

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