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

Weimin Chen

Algebraic & Geometric Topology 15 (2015) 3253–3303

This paper is concerned with fixed-point free S1–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) S1–action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free S1–actions under some further conditions on the fundamental group. The connection between the classification of the S1–manifolds and the fundamental group is given by a certain decomposition, called a fiber-sum decomposition, of the S1–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 S1–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.

four-manifolds, circle actions, Rips–Sela theory, geometrization of $3$–orbifolds
Mathematical Subject Classification 2010
Primary: 57S15
Secondary: 57M07, 57M50
Received: 26 March 2014
Revised: 10 January 2015
Accepted: 20 May 2015
Published: 12 January 2016
Weimin Chen
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA 01003