Surface-fibrations, four-manifolds, and symplectic Floer homology Timothy Perutz Imperial College, University of London A thesis presented for the degree of Doctor of Philosophy of the University of London 2005 Abstract It is known that any smooth, closed, oriented four-manifold which is not negative-definite supports, after blowing up a finite set of points, a ‘broken fibration’. This is a singular surface-fibration of a special kind; its critical points are isolated points and circles.We construct an invariant of broken fibrations and begin a programme to study its properties. It has the same format as the Seiberg-Witten invariant of the underlying four-manifold, and is equal to it in some cases; we conjecture that this is always so. The invariant fits into a field theory for cobordisms equipped with broken fibrations over surfaces with boundary. The formal structure of this theory is very similar to that of a version of monopole Floer ho- mology. The construction is purely symplectic and generalises that of the Donaldson-Smith standard surface count for Lefschetz fibrations. It involves moduli spaces of pseudoholomor- phic sections of relative Hilbert schemes of points on the fibres, with Lagrangian boundary conditions which arise by a vanishing-cycle construction. Broken fibrations form one part of what may be termed ‘near-symplectic geometry’. The other part concerns ‘near-symplectic forms’. The first chapter of the thesis reviews and develops the properties of these forms and their relation with broken fibrations. Declaration The material presented in this thesis is the author’s own, except where it appears with attribution to others. 2 Contents Introduction 5 Geometric structures on four-manifolds . 5 Near-symplectic topology, gauge theory, and pseudoholomorphic curves . 9 1 Near-symplectic geometry 1 1.1 Near-symplectic forms . 1 1.1.1 Topology associated with near-symplectic forms . 13 1.2 Near-symplectic broken fibrations . 23 2 Symmetric products and locally Hamiltonian fibrations 34 2.1 Relative symmetric products . 34 2.1.1 Cohomology of relative symmetric products . 36 2.2 Locally Hamiltonian fibrations . 38 2.3 The vortex equations . 41 2.3.1 Moduli spaces of vortices . 41 2.3.2 The K¨ahlerclass on the vortex moduli space . 46 2.3.3 Families of vortex moduli spaces . 51 3 Invariants from pseudoholomorphic sections 58 3.1 Pseudoholomorphic sections and Lagrangian boundary conditions . 59 3.1.1 Transversality . 61 3.1.2 Compactness . 64 3.1.3 Orientations . 67 3.1.4 Invariance . 70 3.2 Field theories . 70 3.2.1 The theory in outline . 72 3.2.2 Floer homology in the fibre-monotone case . 78 3.2.3 Canonical relative gradings . 80 3 3.2.4 Quantum cap product . 81 3.2.5 Open-closed invariants . 81 3.2.6 Weakly monotone fibres . 85 4 Lagrangian matching invariants 89 4.1 Floer homology for relative symmetric products . 90 4.2 Lagrangian matching invariants . 98 4.2.1 Defining the invariants . 103 4.2.2 Some simple calculations . 107 4.3 Geometry of the compactified families . 111 4.3.1 The Jacobian of a compact nodal curve . 112 4.3.2 Compactifying the Picard family . 114 4.3.3 The Hilbert scheme of r points on a nodal curve . 117 4.3.4 The relative Hilbert scheme . 121 4.4 Constructing the Lagrangian boundary conditions . 125 4.4.1 Vanishing cycles as Lagrangian correspondences . 125 4.4.2 Some auxiliary spaces . 127 4.5 Floer homology of the Jacobian families . 133 4.6 An exact triangle? . 135 A Symmetric products of non-singular complex curves 138 B Topology of mapping tori 141 4 Introduction Geometric structures on four-manifolds Two-forms and almost complex structures A possible approach to understanding the four-dimensional smooth category, with its enig- mas, is to begin with a restricted class of manifolds, sufficiently tractable that at least some of the mysteries can be resolved, and then progressively to enlarge the class. A good way to do this is via triples (X, ω, J), where the closed four-manifold X has been equipped with a closed two-form ω Z2 and an almost complex structure J End(TX). The first class ∈ X ∈ to consider is of triples which make X a compact K¨ahlersurface. General four-manifolds do not support almost complex structures, and we shall consider situations where J is defined only on an dense open subset U X. We always insist that, ⊂ where it is defined, it tames ω: ωx(v, Jv) > 0 for every non-zero v TxU, ∈ so that ω U is symplectic. Usually we also impose the compatibility relation | ωx(Jv, v0) + ωx(v, Jv0) = 0, which means that g := ω( ,J ) is a Riemannian metric. Then (U, ω U, J) is, by definition, ∙ ∙ | an almost K¨ahlermanifold. Global almost K¨ahlerstructures are much more plentiful than the K¨ahlerones, where J is required to be integrable and ω of type (1, 1); many compact symplectic four-manifolds, even simply connected ones, do not support integrable complex structures. However, the property of supporting a symplectic structure is an even subtler one (to give a simple example, a necessary but insufficient condition is that, in any connected + sum decomposition, one of the summands has b2 = 0). We can relax the almost-K¨ahlercondition, allowing ω to be a near-symplectic form. 2 This means that at each point x either ωx > 0 or ωx = 0, and that along its zero-locus 1 2 Zω = ω− (0), ω satisfies the transversality condition that xω : TxX Λ T ∗X has rank ∇ → x 5 3 (which is the largest possible). This implies that Zω is an embedded 1-submanifold. We then ask that (X Zω, ω (X Zω),J) be almost K¨ahler,and that the associated metric g \ | \ extends to a metric on X (J will then not extend). It has been known for several years that near-symplectic structures (X, ω, J), compatible with a chosen orientation of X, exist if and only if there exists a class c H2(X; R) with ∈ c2[X] > 0. This is a consequence of Hodge theory, together with a non-trivial transversality argument. If cg is the harmonic representative of such a c, for the Riemannian metric g, + then its self-dual part ωc,g = cg is a near-symplectic form for generic g. Pencils and fibrations A topological Lefschetz pencil (X, B, π) on X is given by a discrete subset B X and ⊂ a smooth map π : X B S to a smooth oriented surface S such that the fibres of π have \ → compact closures in X. Each point b B must have a neighbourhood Ub such that the map ∈ Ub b π(Ub b ) is equivalent, as a germ of maps of oriented manifolds, to the map 2 \{ } → 1 \{ } C 0 CP ,(z1, z2) (z1 : z2). Each point p X B has a neighbourhood Up such \{ } → 7→ ∈ \ 2 that π :(Up, p) (π(Up), π(p)) is equivalent to one of two maps (C , 0) (C, 0)—either → → (z1, z2) z1z2, or (z1, z2) z1. It is a topological Lefschetz fibration when B = . 7→ 7→ ∅ There is a simple device which converts a Lefschetz pencil over S2 into a Lefschetz fibration: one blows up X along B (using local complex coordinates near B as in the model above) and composes π with the blow-down map σ : X X. This has a unique extension → to a Lefschetz fibration π σ : X S2, the exceptional divisors appearing as sections. ◦ → In the preprint [3], Auroux, Donaldson and Katzarkovb introduce the following general- isation of the notion of topologicalb Lefschetz pencil. I have adopted Ivan Smith’s punning suggestion for its name. Definition 0.0.1. A broken pencil (X, B, π) consists of a smooth oriented 4-manifold X, a 0-submanifold B X and a smooth map π : X B S to an oriented surface, such that ⊂ \ → The fibres of π have compact closures in X. • The critical set crit(π) X B is a disjoint union of a 1-submanifold Z and a • ⊂ \ discrete set of points D. Each point b B must has an open neighbourhood Ub such ∈ that the map Ub b π(Ub b ) is equivalent, as a germ of maps of oriented \{ } →2 \{ }1 manifolds, to the map C 0 CP ,(z1, z2) (z1 : z2). Each point p D has an \{ } → 7→ ∈ open neighbourhood Up such that π :(Up, p) (π(Up), π(p)) is equivalent to the map 2 → (C , 0) (C, 0), (z1, z2) z1z2. → 7→ For any component Zi Z, π Zi is injective; and for any other component Zj, either • ⊂ | π(Zi) = π(Zj) or π(Zi) π(Zj ) = . ∩ ∅ 6 Each point p Z has a neighbourhood Up such that π :(Up, p) (π(Up), π(p)) is • ∈ → equivalent to a map 3 2 R R R , (x, t) (q(x), t), × → 7→ where q : R3 R is a non-degenerate quadratic form of signature (2, 1). → A broken fibration is a pair (X, π) such that (X, , π) is a broken pencil. ∅ In the above, ‘equivalence’ is of smooth germs, respecting the orientations of X and S. So the last point means that there is an orientation-preserving diffeomorphism Ψ from Up to an open subset of R4, with Ψ(p) = 0, and an orientation-preserving diffeomorphism ψ from an open subset neighbourhood of π(p) in S to an open subset of R2, with ψ(π(p)) = 0, 1 4 such that the equation ψ f Ψ− (x, t) = (q(x), t) holds on some neighbourhood of 0 R .