Fixed-point free circle actions on 4–manifolds

Abstract

Abstract. This paper is concerned with fixed-point free S 1 -actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) S 1 -action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free S 1 -actions under some further conditions on the fundamental group. The connection between the classification of the S 1 -manifolds and the fundamental group is given by a certain decomposition, called fiber-sum decomposition, of the S 1 -manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z-splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z-splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela [39]. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free S 1 -action on the 4-manifold must be infinite, unless the 4-manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3-manifold.