#### Volume 15, issue 6 (2015)

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Fixed-point free circle actions on $4$–manifolds

### Weimin Chen

Algebraic & Geometric Topology 15 (2015) 3253–3303
##### Abstract

This paper is concerned with fixed-point free ${\mathbb{S}}^{1}$–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) ${\mathbb{S}}^{1}$–action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free ${\mathbb{S}}^{1}$–actions under some further conditions on the fundamental group. The connection between the classification of the ${\mathbb{S}}^{1}$–manifolds and the fundamental group is given by a certain decomposition, called a fiber-sum decomposition, of the ${\mathbb{S}}^{1}$–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. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free ${\mathbb{S}}^{1}$–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.

##### Keywords
four-manifolds, circle actions, Rips–Sela theory, geometrization of $3$–orbifolds
##### Mathematical Subject Classification 2010
Primary: 57S15
Secondary: 57M07, 57M50