Symplectic 4-manifolds with fixed point free circle actions

We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds that admit non-trivial circle actions.


Introduction
Recently Friedl and Vidussi, [7] solved the long standing Taubes Conjecture, which classifies which 4-manifolds of the form M × S 1 admit symplectic forms. Moreover, they determined exactly which cohomology classes can be represented by symplectic forms. Using recent results of D. Wise, [13] they have extended their results to the case of non-trivial S 1 -bundles in [9]. In this note we observe that their results as well of those of Chen, who obtained partial results in the fixed point free case in [3], imply the analogue of ( [9], Theorem 1.3) for all fixed point free circle actions.
Before stating our main result we fix some notation and terminology. Let X p −→ M be an orientable 4-manifold with a fixed point free circle action and quotient space M = X/S 1 . The quotient space is an orbifold whose underlying topological space |M| is a manifold since all the stabilisers of the S 1 -action are necessarily cyclic and the singular locus consists of a collection of branching circles (cf. [1], [6]).
We let M reg denote the complement of an open tubular neighbourhood of the singular locus of M and X reg = p −1 (M reg ), which is an honest S 1 -bundle so that the pushforward map p * is well-defined for cohomology classes in H * (X reg , R). The manifold M reg has toroidal boundary and thus one may define the Thurston norm on H 1 (M reg , R) in the usual fashion. Finally for ψ ∈ H 2 (X, R) we let ψ reg denote the restriction of ψ to X reg . Theorem 1. Let X p −→ M be an oriented manifold admitting a fixed point free S 1 -action with quotient space M and let ψ ∈ H 2 (X, R). Then the following are equivalent: (1) ψ can be represented by a symplectic form, is required to be transverse to the boundary. Furthermore, since φ reg is the restriction of a class in H 1 (|M|, R), it automatically vanishes on the meridian classes in ∂M reg so that if φ reg is fibered, then the induced fibration on the boundary is necessarily meridional. Thus the fibration dual to φ reg extends to |M| in the desired way by filling in discs near the singular locus. In particular, part (3) of Theorem 1 implies that the underlying manifold |M| is fibered and we obtain a positive answer to the following conjecture, which implies ( [3], Conjecture 1.7) as a special case.
Conjecture 1 (Generalised Taubes conjecture). Let X be a symplectic 4-manifold that admits a non-trivial fixed point free circle action with quotient orbifold M. Then the (possibly empty) singular locus L of M is a meridionally fibered link.
Furthermore, as noted in ( [3], p. 6), Theorem 1 completes the characterisation of which symplectic manifolds admit non-trivial S 1 -actions. For Baldridge, [1] showed that if a nontrivial S 1 -action on a symplectic 4-manifold has fixed points then X is rational or ruled and thus admits an S 1 -invariant symplectic form for some non-trivial S 1 -action. In view of this we obtain the following corollary.
Corollary 1. Let X be a symplectic 4-manifold that admits a non-trivial S 1 -action. Then either the action is fixed point free and the quotient space fibers over S 1 or X is rational or ruled. In either case, X admits a non-trivial symplectic S 1 -action.

Proof of Theorem 1
The proof of Theorem 1 is based on the following lemma, which provides a generalisation of ([4], Theorem 5.2) to include irrational classes. For the proof we assume a certain familiarity with the basic properties of the Thurston norm (cf. [12]). Proof. We first assume that φ is rational. Since nothing changes after multiplying with positive constants, we may assume that φ is in fact integral. In this case the first claim is just a restatement of ([4], Theorem 5.2), which can be applied in complete generality in view of ( [11], Theorem 8.1). Note that the assumption H 1 (M , Q) G = Q in ( [4], Theorem 5.2) can be replaced by the fact that the fibration is given by a fibered class φ that is G-invariant. Moreover, the proof in [4] actually gives a fibration that is transverse to the branching locus in M . The quotient map π induces an isomorphism H 1 (M , R) G ∼ = H 1 (|M|, R) so that there is a unique class φ with φ = π * φ and the fibration dual to φ descends to a fibration of |M| dual to φ. Finally since the fibration is transverse to the singular locus it follows that the restriction of φ to M reg is fibered.
We next assume that φ is irrational and let φ be the unique class with φ = π * φ. We let ι reg denote the natural inclusion M reg ֒→ |M| and set V = Im(ι * reg ). By the previous case all rational classes in V that are sufficiently close to φ reg = ι * reg φ are fibered. If φ reg itself did not lie in the open cone over a fibered face of the Thurston unit ball, then it must lie in the closed cone over the boundary of a fibered face by the assumption that it can be approximated by fibered elements. Since the Thurston unit ball is rational, the intersection of the closed cone containing φ reg with V must contain non-fibered rational points arbitrarily close to φ reg , which gives a contradiction. Thus φ reg admits a non-degenerate de Rham representative η reg . Since η reg can be approximated by rational classes that are fibered and restrict to meridional fibrations on the boundary of M reg , the foliation induced by η reg on the boundary is also meridional.
We let (z, θ) ∈ D 2 × S 1 denote coordinates on a tubular neighbourhood of a component of the branching locus of |M|. After applying a suitable isotopy we may assume that η reg has the form f (θ)dθ near ∂D 2 × S 1 . It follows that η reg extends to a non-degenerate closed form η which is transverse to the branching locus of |M|. The pullback η = π * η then gives the desired non-degenerate G-equivariant representative of φ.
(1) =⇒ (3): Let (X, ω) be a symplectic manifold with a fixed point free S 1 -action and quotient space M. By ([3], Proposition 1.8) there is a manifold M and a smooth action by a finite group so that M = M /G. Furthermore, we have the following commutative diagram: where π is the quotient map, π is an unramified covering and the induced S 1 -action on X is free. Moreover, the group G acts naturally on X as the group of deck transformations of π.
Thus ω = π * ω is a symplectic form and by ([9], Theorem 1.4) its image under the pushforward map p * (ω) ∈ H 1 (M , R) lies in the open cone over a fibered face of the Thurston norm ball. Since ω is G-invariant and the action on X is fiber preserving, the class φ = p * (ω) is also G-invariant. We let φ be the unique class such that π * φ = φ. By Lemma 1 the restriction φ reg to M reg lies in the open cone over a fibered face. Finally the naturality of the transfer homomorphism implies that the restriction of φ reg agrees with p * ω reg .
(3) =⇒ (2): By assumption φ reg = p * ψ reg lies in the open cone over a fibered face of the Thurston norm ball and φ reg is the restriction of a class φ ∈ H 1 (|M|, R). In particular, |M| fibers over S 1 . We first note that M is a very good orbifold so that it is a quotient of a manifold M by a smooth action of a finite group G. For this it suffices to rule out bad 2-suborbifolds by ( [2], Corollary 3.28). However, a bad 2-suborbifold is topologically a sphere that is essential in H 2 (|M|, Z) and as in the proof of ([3], Lemma 2.3) this implies that b + 2 (X) = b 2 (|M|) − 1. Thus |M| = S 2 × S 1 and b + 2 (X) = 0, contradicting the assumption that ψ 2 > 0.
Thus since M is very good we can proceed as in the proof of the previous implication. In particular, M is a quotient of a manifold M by a smooth action of a finite group G, the total space has a finite covering X which is a genuine S 1 -bundle and these bundles fit into a pullback diagram as above. Since a degree one cohomology class on M is determined by its restriction to the complement of the branching locus, we deduce that φ = π * φ and p * (π * ψ) agree as cohomology classes. We then note that the construction of S 1 -invariant forms in [5] and its extension to irrational classes ( [8], Theorem 1.1) can be done G-equivariantly.
First choose a G-invariant representative γ of e(X), which can be obtained as the curvature of a G-equivariant angular form. By ([8], Lemma 2.1) we may write γ = φ ∧ β. After averaging over G this equation still holds, so β can be assumed to be G-equivariant. Let η be a G-invariant angular 1-form so that dη = p * γ and let Ω ∈ H 2 (M , R) be the unique class such that the following holds in cohomology Such an Ω exists in view of the Gysin sequence since the left hand lies in the kernel of p * and since the left hand side is G-equivariant so is Ω. The fact that π * ψ 2 > 0 implies that p * φ ∧ Ω > 0. Thus by ([8], Lemma 2.2) there is a non-vanishing 2-form representing the class Ω so that φ ∧ Ω > 0, again after averaging we may assume that Ω is G-invariant. Thus the S 1 -invariant form ω inv = η ∧ p * φ + p * Ω represents π * ψ and descends to an S 1 -invariant form ω inv on X which is cohomologous to ψ. Remark 1. A vital step in the proof of Theorem 1 was Chen's observation that the base orbifold of a symplectic manifold X admitting a fixed point free S 1 -action is a quotient of a manifold by a finite group action. The main technical point in the proof of ([3], Proposition 1.8) is to rule out bad 2-orbifolds in the base. This is achieved by results relating the Seiberg-Witten invariants of the base orbifold to those of the underlying manifold.
We sketch a different proof which uses more standard Seiberg-Witten vanishing results. For background on the Seiberg-Witten invariants we refer to [10] and the references therein. First observe that a bad 2-suborbifold Σ in the quotient orbifold M = X/S 1 can intersect at most 2 singular curves L 1 , L 2 each in at most one point. Taking a neighbourhood N of |Σ| ∪ L 1 gives a topological splitting of the base |M| = (S 2 × S 1 )#M ′ so that preimage of the splitting sphere in |M| induces a splitting X = X 1 ∪ S X 2 , where S is either S 2 × S 1 or S 3 depending on whether L 2 is empty or not. Moreover, as in the proof of ([3], Lemma 2.3) we must have b 1 (M ′ ) > 0 by the assumption that b + 2 (X) > 0. If S is a 3-sphere, then b + 2 (X 2 ) > 0 and by taking the covering X of X induced by the natural surjection π 1 (X) → π 1 (S 2 × S 1 ) → Z n we obtain a splitting of X = X 1 ∪ S 3 X 2 , where b + 2 (X 1 ), b + 2 (X 2 ) ≥ 1. It follows that the Seiberg-Witten invariants of X are trivial.
The embedded 2-sphere S 2 × {pt} in S then becomes essential in the covering and b 1 (X), and hence b + 2 (X), may be assumed to be arbitrarily large. Furthermore, the sphere S 2 ×{pt} has trivial self-intersection and consequently the Seiberg-Witten invariants of X are trivial. Thus in both cases we obtain a contradiction to the non-vanishing results of Taubes for the Seiberg-Witten invariants of a symplectic 4-manifold.