Invariants in Contact Topology

Yakov Eliashberg^{1}

Abstract. Contact topology studies contact manifolds and their Legendrian submanifolds up to contact diffeomorphisms. It was born, together with its sister Symplectic topology, less than 20 years ago, essentially in seminal works of D. Bennequin and M. Gromov ( see [2, 18]). However, despite several remarkable successes the development of Contact topology is still significantly behind its symplectic counterpart. In this talk we will discuss the state of the art and some recent breakthroughs in this area.

1991 Mathematics Subject Classification: 53C, 55N, 57R, 58G, 81T Keywords and Phrases: Contact manifolds, Legendrian submanifolds, holomorphic curves, contact homology algebra

1 Contact preliminaries

A 1-form α on a (2n − 1)-dimensional manifold V is called contact if the restriction of dα to the (2n − 2)-dimensional tangent distribution ξ = {α = 0} is non-degenerate (and hence symplectic). A codimension 1 tangent distribution ξ on V is called a contact structure if it can be locally (and in the co-orientable case globally) defined by the Pfaffian equation α = 0 for some choice of a contact form α. The pair (V, ξ) is called a contact manifold. According to Frobenius’ theorem the contact condition is a condition of maximal non-integrability of the tangent hyperplane field ξ. In particular, all integral submanifolds of ξ have dimension ≤ n − 1. On the other hand, (n − 1)-dimensional integral submanifolds, called Legendrian, always exist in abundance. Any non-coorientable contact structure can be canonically double-covered by a coorientable one. If a contact form α is fixed then one can associate with it the Reeb vector field R_{α}, which is transversal to the contact structure ξ = {α = 0}. The field R_{α} is uniquely determined by the equations R_{α} dα = 0; α(R_{α}) = 1 .

The 2n-dimensional manifold M = (T (V)/ξ)^{∗} \ V , called the symplectization of (V, ξ), carries the natural symplectic structure ω induced by the embedding M → T^{∗}(V) which assigns to each linear form T (V)/ξ → ℝ the corresponding form T (V) → T (V)/ξ → ℝ. A choice of a contact form α (if ξ is co-orientable) defines a splitting M = V × (ℝ \ 0). We will usually pick the positive half V × ℝ_{+} of M, and call it symplectization as well. The symplectic structure ω can be written in terms of this splitting as d(τα), τ > 0. It will be more convenient for us, however, to use additive notations and write ω as d(e^{t}α), t ∈ ℝ, on M = V × ℝ. Notice that the vector field T = \frac{∂}{∂t} is conformally symplectic: we have L_{T} ω = ω, as well as L_{T} (e^{t}α) = e^{t}α. All the notions of contact geometry can be formulated as the corresponding symplectic notions, invariant or equivariant with respect to this conformal action. For instance, any contact diffeomorphism of V lifts to an equivariant symplectomorphism of M; contact vector fields on V (i.e. vector fields preserving the contact structure) are projections of ℝ-invariant contact symplectic (and automatically Hamiltonian) vector fields on M; Legendrian submanifolds in M correspond to cylindrical (i.e. invariant with respect to the ℝ-action) Lagrangian submanifolds of M.

1Supported by the National Science Foundation

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The symplectization of a contact manifold is an example of a symplectic manifold with cylindrical (or rather conical) ends, which is a possibly noncompact symplectic manifold (W, ω) with ends of the form E_{+} = V_{+} × [0, ∞) and E_{−} = V_{−} × (−∞, 0], such that V_{±} are compact manifolds, and ω|_{V_{±}} = d(e^{t}α_{±}), where α_{±} are a contact forms on V_{±}. In other words, the ends E_{±} of (W, ω) are symplectomorphic, respectively, to the positive, or negative halves of the symplectizations of the contact manifolds (V_{±}, ξ_{±} = {α_{±} = 0}). We will consider the splitting of the ends and the the contact forms α_{±} to be parts of the structure of a symplectic manifold with cylindrical ends. We will also call (W, ω) a directed symplectic cobordism between the contact manifolds (V_{+}, ξ_{+}) and (V_{−}, ξ_{−}), and denote it, sometimes, by \overrightarrow{V_{+}V_{−}}. Let us point out that this is not an equivalence relation, but rather a partial order. Existence of a directed symplectic cobordism \overrightarrow{M_{+}M_{−}} does not imply the existence of a directed symplectic cobordism \overrightarrow{M_{−}M_{+}}, but directed symplectic cobordisms \overrightarrow{M_{0}M_{1}} and \overrightarrow{M_{1}M_{2}} can be glued, in an obvious way, into a directed symplectic cobordism \overrightarrow{M_{0}M_{2}}. Suppose now that the symplectic form ω is exact and equal dβ, where β_{|E_{±}} = e^{t}α_{±}, and that there exists a Morse function φ : W → ℝ which coincides with the function t at infinity and such that for any c ∈ ℝ the restriction β|_{{φ=c}} is a contact form away from the critical points of the function φ. In this case we say that (W, ω) is a directed Stein cobordism between the contact manifolds (V_{+}, ξ_{+}) and (V_{−}, ξ_{−}). Notice that indices of critical points of the function φ are bounded in this case by n = \frac{1}{2}dimW. If there exists a directed symplectic (resp. Stein) cobordism between a contact manifold (V_{+}, ξ_{+}) and V_{−} = ∅, then (V_{+}, ξ_{+}) is called symplectically (resp. Stein) fillable.^{2} The Stein filling W is called subcritical if the function φ can be chosen without critical points of the maximal index n.

Contact structures have no local invariants. Moreover, any contact form is

locally isomorphic to the form α_{0} = dz − ∑_{1}^{n−1} y_{i}dx_{i} (Darboux’ normal form). The

contact structure ξ_{0} on ℝ^{2n−1} given by the form α_{0} is called standard. Standard contact structure on S^{2n−1} is formed by complex tangent hyperplanes to the unit sphere in ℂ^{n}. The standard contact structure on S^{2n−1} is isomorphic in the complement of a point to the standard contact structure on ℝ^{2n−1}. According to a theorem of J. Gray (see [17]) contact structures on closed manifolds have the following stability property: Given a family ξ_{t}, t ∈ [0, 1], of contact structures on a closed manifold M, there exists an isotopy f_{t} : M → M, such that df_{t}(ξ_{0}) = ξ_{t}; t ∈ [0, 1]. Notice that for contact forms the analogous statement is wrong. For instance, the topology of the 1-dimensional foliation determined by the Reeb vector field R_{α} is very sensitive to deformations of the contact form α.

2The Stein fillabillity of (V, ξ) is equivalent to the existence of a compact complex manifold with a strictly pseudoconvex boundary V and a Stein interior, such that ξ is the field of complex tangencies to the boundary V. See [9] for the discussion of different notions of symplectic fillabillity.

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The conformal class of the symplectic form dα|_{ξ} depends only on the cooriented contact structure ξ and not a choice of the contact form α. In particular, one can associate with ξ an almost complex structure J : ξ → ξ, compatible with dα which means that dα(X, JY); X, Y ∈ ξ, is an Hermitian metric on ξ. The space of almost complex structures J with this property is contractible, and hence the choice of J is homotopically canonical. Thus a cooriented contact structure ξ defines on M a stable almost complex structure J̃ = J̃_{ξ}, i.e. a splitting of the tangent bundle T (V) into the Whitney sum of a complex bundle of (complex) dimension (n − 1) and a trivial 1-dimensional real bundle. The existence of a stable almost complex structure is necessary for the existence of a contact structure on V. If V is open (see [19]) or dimV = 3 (see [25, 24]) this property is also sufficient for the existence of a contact structure in the prescribed homotopy class. It is still unknown whether this condition is sufficient for the existence of a contact structure on a closed manifold of dimension > 3. However, the positive answer to this question is extremely unlikely. Similarly, the homotopy class of J̃_{ξ}, which we denote by [ξ] and call the formal homotopy class of ξ, serves as an invariant of ξ. For an open V it is a complete invariant (see [19]) up to a deformation of contact structures, but not up to a contact diffeomorphism. For closed manifolds this is known to be false in all dimensions, see the discussion below. The main goal of this talk is the construction of invariants which would allow to distinguish contact manifolds in the same formal homotopy class.

2 Invariants of open manifolds

We concentrate in this section on 3-dimensional contact manifolds, although some part of the discussion can be generalized to higher dimensions. First of all 3dimensional contact manifolds are orientable, and any contact structure determines an orientation of M. If M is a priori oriented then contact structures can be divided into positive and negative. We will consider here only positive contact structures.

It is proven to be useful to divide all 3-dimensional contact manifolds into two complementary classes: tight and overtwisted. A contact 3-manifold (M, ξ) is called overtwisted if there exists an embedded disc D^{2} ⊂ M such that its boundary ∂D^{2} is tangent to ξ (i.e. ∂D^{2} is a Legendrian curve), while the disc itself is transverse to ξ along its boundary. A non-overtwisted contact structures are called tight. D. Bennequin (see [2]) was the first who discovered the phenomenon of overtwisting. He proved that the standard contact structure ξ_{0} on S^{3} is tight and constructed an overtwisted contact structure ξ_{1} in the same formal homotopy class.

As it turned out, overtwisted contact structures on all closed 3-manifolds are classified up to isotopy by their formal homotopy classes (see [7]). On open manifolds one should subdivide furthermore overtwisted contact structures into overtwisted at infinity and tight at infinity. A contact structure, which is overtwisted at infinity, is determined up to isotopy by its formal homotopy class (see [6]). Overtwisted, but tight at infinity contact structure on M \ F, where M is a closed 3-manifold and F is its finite subset, can always be canonically extended to M see [6]), and thus an isotopical classification of such structures coincides with the formal homotopical classification on M (and not on M \ F !).

Let us now restrict ourself to the class of tight contact structures. It is not so easy to provide non-trivial invariants of tight contact structures on open manifolds. The problem is that all standard symplectic invariants (the Gromov width, capacities, etc.) take infinite values for symplectizations of contact manifolds. One knows, for instance, that on ℝ^{3} any tight contact structure is isotopic to the standard one (see [6]). However, on the closed half-space ℝ_{+}^{3} = {y ≥ 0} ⊂ ℝ^{3} there are non-isomorphic contact structures, which we are describing below. Let us denote by Ξ_{0} the space of tight contact structures on ℝ_{+}^{3} which coincide with the standard contact structure ξ_{0} near the plane Π = {y = 0} = ∂ℝ_{+}^{3}. We are interested in invariants of contact half-spaces (ℝ_{+}^{3}, ξ), where ξ ∈ Ξ_{0}, up to diffeomorphisms fixed near Π.

Given a contact structure ξ ∈ Ξ_{0} let us consider an embedded plane Π̃ ⊂ ℝ_{+}^{3} which coincides with Π at infinity, and which is transversal to ξ_{0}. The 1dimensional line field ξ ∩ T (Π̃) integrates into a 1-dimensional characteristic foliation Π̃_{ξ} on Π̃. The foliation Π̃_{ξ} coincides with the foliation by lines {z = const} at infinity, and thus the holonomy along its leaves defines a compactly supported diffeomorphism hΠ̃ : ℝ → ℝ, where we identify the source ℝ with the line {x = −N} ⊂ Π, and the target ℝ with the line {x = N} ⊂ Π for a sufficiently large N > 0. Let us define c_{ξ}(z) = sup ; Π̃ (hΠ̃(z) − z) , and call the function c_{ξ}(z)

the contact shape of (ℝ^{3}, ξ). Of course, sometimes we have c(ξ) ≡ +∞. For instance this is the case for the standard contact structure ξ = ξ_{0}. On the other hand, the following construction (see [12]) shows that any positive continuous ^{3} function f : ℝ → ℝ, such that f(z) + z is a monotone function, can be realized as the invariant c_{ξ} for some contact structure ξ ∈ Ξ_{0}.

For a positive Lipschitz function φ on ℝ^{2} we denote by S_{φ} its graph {y = φ(x, z)} ⊂ ℝ^{3}. If the function φ decays sufficiently fast when |x| → ∞ (say, φ(x, z) < \frac{C}{x^{2}} ), then the holonomy diffeomorphism h_{φ} : ℝ → ℝ along the leaves of the characteristic foliation of the graph Π_{φ} = {y = φ(x, z)} is well defined. It is easy to find a Lipshitz function φ with the prescribed continuous holonomy h_{φ}(z) = z + f(z). Consider the domain Ω_{φ} = {0 ≤ y < φ(x, z)}. Clearly, the contact manifold (Ω_{φ}, ξ_{0}) belongs to the class Ξ_{0}. We have (see [12])

Proposition 2.1

c_{(Ω_{φ},ξ_{0})}(z) = h_{φ}(z) − z = f(z) .

A similar invariant can be defined for open manifolds (without boundary) when H_{1}(M) ≠ 0. (see [5])

3In fact, it need not to be even continuous.

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3 Invariants of closed manifolds

Until very recently there was known only one result allowing to distinguish contact structures on manifolds of dimension > 3 within a given formal homotopy class. Namely, we have

Theorem 3.1 For any n > 2 the sphere S^{2n−1} has a contact structure ξ_{1} in the standard formal homotopy class, which is not isomorphic to the standard contact structure ξ_{0}.

For odd values of n this was shown by the author in [4], and later extended to even values of n by H. Geiges (see [15]). ^{4}

We will discuss in this section some new powerful algebraic invariants of closed contact manifolds, related to Gromov-Witten invariants of symplectic manifolds, which were recently developed jointly by H. Hofer, A. Givental and the author. See also Hofer’s talk at the current proceedings for the discussion of other related aspects of this theory.

Contact homology algebra. To define the invariants of a contact manifold (V, ξ) let us fix a contact form α and an almost complex structure J : ξ → ξ compatible with the symplectic form dα. The symplectization M of (V, ξ) can be identified, as was explained in Section 1, with (V × ℝ, d(e^{t}α)). The complex structure J extends from ξ to T (M) by setting J \frac{∂}{∂t} = R_{α}, where R_{α} is the Reeb vector field of the contact form α. For a generic choice of α there are only countably many periodic trajectories (including multiple ones) of the vector field R_{α}. Moreover, these trajectories can be assumed non-degenerate which means that the linearized Poincaré return map along any of these trajectories has no eigenvalues equal to 1.

Let P = P_{α} be the set of all periodic trajectories of R_{α}. We do not fix initial points on periodic trajectories, and include all multiples as separate points of P. Let us first assume that H_{1}(V) = 0. For each γ ∈ P let us choose and fix a surface F_{γ} spanning the trajectory γ in V. This enable us to define the Conley-Zehnder index μ(g) of γ as follows. Choose a homotopically unique trivialization of the symplectic vector bundle (ξ, dα) over each trajectory γ ∈ P which extends to ξ|_{F_{γ}}. The linearized flow of R_{α} along γ defines then a path in the group Sp(2n−2, ℝ) of symplectic matrices, which begins at the unit matrix and ends at a matrix with all eingenvalues different from 1. The Maslov index of this path (see [1, 26]) is, by the definition, the Conley-Zehnder index μ(γ) of the trajectory γ. For our purposes it will be convenient to use the reduced Conley-Zehnder index \overline{γ} = μ(γ) + n − 3, also called the degree of γ. Notice that by changing the spanning surfaces for the trajectories from P one can change Conley-Zehnder indices by the values of the cohomology class 2c_{1}(ξ), where c_{1}(ξ) is the first Chern class of the contact bundle ξ. In particular, mod 2 indices can be defined independently of any spanning surfaces, and even in the case when H_{1}(V) ≠ 0.

4Although this result sounds similar to Bennequin’s theorem asserting that the standard contact structure on S^{3} is not overtwisted, non-standard contact structures on high-dimensional spheres provided by Theorem 3.1 are quite different: they are symplectically, and even Stein fillable, while an overtwisted contact structure on S^{3} is not.

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Next, we consider certain moduli spaces of holomorphic curves in the manifold M = V × ℝ, which are essential for all our algebraic constructions. Let us observe that for each periodic orbit γ ∈ P the cylinder γ ×ℝ is a J-holomorphic curve. Let D_{r} be the disc of radius r in ℂ centered at the origin. Given any orbit γ ∈ P we say that a J-holomorphic map f : D_{r}\0 → M = V ×ℝ converges near 0 to the periodic trajectory γ at ±∞ if f(z) = (g(z), h(z)), h(z) →_{|z|→0} = ±∞, and there exits the

limit ḡ(φ) = lim_{ρ→0} g(ρe^{iφ}) which parametrizes the periodic trajectory γ. Notice that

the orientation which is defined this way on γ coincides with the orientation given by the Reeb vector field ℝ_{α} at +∞, and opposite to this orientation at −∞.

Let us denote by S_{sr}, s, r = 0, 1, …, the 2-sphere S^{2} with s+r fixed punctures y_{1}, …, y_{s}, x_{1}, …, x_{r}. Given s + r periodic orbits γ_{1}, …, γ_{s}, δ_{1}, …, δ_{r} ∈ P we consider the space M̃^{A}(γ_{1}, …, γ_{s}; δ_{1}, …, δ_{r}), which consists of pairs (f, j), where j is a conformal structure on S_{sr}, and f : S_{sr} → M is a (j, J)-holomorphic curve, such that near each puncture y_{k}, k = 1, …, s, the map f converges to γ_{k} at +∞, and near each puncture x_{l}, l = 1, …, r, it converges to δ_{l} at −∞. As usual we pass to the corresponding moduli space M(γ_{1}, …, γ_{s}; δ_{1}, …, δ_{r}) by identifying pairs (f, j) and (f̃, j̃) which differ by a diffeomorphism of the sphere S^{2} which fixes the punctures y_{1}, …, y_{s}, x_{1}, …, x_{r}. The space M can be written as a disjoint union M = ⋃_{A∈H_{2}(V)} M^{A}, where M^{A} consists of holomorphic curves which together with

the surfaces spanning in V the trajectories γ_{1}, …, γ_{s}, δ_{1}, …, δ_{r} ∈ P represent the homology class A ∈ H_{2}(M) = H_{2}(V). Then we have

Proposition 3.2 For a generic choice of J, and any periodic orbits γ_{1}, …, γ_{s}, δ_{1}, … δ_{r} ∈ P the moduli space M^{A}(γ_{1}, …, γ_{s}; δ_{1}, … δ_{r}) is an orbifold of dimension ^{5}

∑_{k=1}^{s} \overline{γ_{k}} − ∑_{l=1}^{r} \overline{δ}_{l} + (2 − 2s)(n − 3) + 2c_{1}(ξ)[A].

Remark 3.3 The additive group ℝ acts on M = V ×ℝ by J-biholomorphic translations (x, t) ↦ (x, t+c). The moduli spaces M(γ_{1}, …, γ_{s}; δ_{1}, …, δ_{r}) are invariant under this action, and hence, with the exception of trivial spaces M(γ; γ) (which consist of cylinders γ × ℝ), a non-empty moduli space M(γ_{1}, …, γ_{s}; δ_{1}, …, δ_{r}) always has a positive dimension.

Let us consider now a free (super-)commutative graded algebra Θ = Θ_{α} over ℂ with the unit element generated by elements of P_{α}. In other words, Θ is a polynomial algebra with complex coefficients of generators of even degree and an exterior algebra of odd degree generators. Let us recall that we count all multiples of a given trajectories as independent generators of Θ. Each monomial element θ ∈ Θ is graded by its total degree θ̄. Let Θ^{H_{2}} be the group algebra of H_{2}(V) = H_{2}(M) with coefficients in Θ. Thus elements of Θ^{H_{2}} can be written as

5It is a standard difficulty in the Floer homology theory and the theory of holomorphic curve invariants in general, that in the presence of multiply-covered curve it is, sometimes, impossible to achieve transversality needed for this dimension formula just by perturbing the almost complex structure J. The appropriate virtual cycles technique which works in this case and involves multivalued perturbations was recently developed by several authors, see [14], [23] et al.

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polynomials ∑_{A∈H_{2}(V)} θ_{A}t^{A}, θ_{A} ∈ Θ.

We define now a sequence of operations \underbrace{Θ^{H^{2}} ⊗ … Θ^{H_{2}}}_{s} → Θ^{H_{2}}, which make Θ

into a L_{∞}-algebra (see, for instance, [22]), or rather a P_{∞}-algebra, where P stands for the Poisson structure. This is done by an appropriate counting of components of the moduli spaces M^{A}(γ_{1}, …, γ_{s}; δ_{1}, … δ_{r}).

Take any s ≥ 1 periodic orbits γ_{1}, …, γ_{s} ∈ P and set

[γ_{1}, …, γ_{s}]_{s} = ∑_{A∈H_{2}(V)} ∑_{Δ} a_{Δ}^{A}t^{A}Δ,

where a_{Δ}^{A} ∈ ℂ, and the second sum is taken over all monomials Δ = δ_{1}^{j_{1}} ⋯ δ_{r}^{j_{r}} of (distinct) generators δ_{1}, …, δ_{r}, … ∈ Θ (notice that we allow the case r = 0).

We set a_{Δ}^{A} = 0 if the dimension ∑_{k=1}^{s} \overline{γ_{k}} − ∑_{l=1}^{r} j_{l}\overline{δ_{l}} + (2 − 2s)(n − 3) + 2c_{1}(ξ)[A] of

the moduli space M^{A} = M^{A}(γ_{1}, …, γ_{s}; \underbrace{δ_{1} …, d_{1}}_{j_{1}}, …, \underbrace{d_{r}, …, δ_{r}}_{j_{r}}) is different from

1. Otherwise, we define the coefficient a_{Δ}^{A} as the sum ∑_{C} w(C) of weights w(C)

assigned to 1-dimensional components of the moduli space M^{A} Given a component C of M we set

w(C) = ±\frac{1}{r!d}m(δ_{1})^{j_{1}} ⋯ m(δ_{r})^{j_{r}},

where m(δ_{l}) is the multiplicity of the periodic orbit δ_{l}, l = 1, …, r; d = 1 if the curves from C are not multiply-covered and d is the order of the group of deck transformations of the corresponding branched covering in the multiply-covered case. Finally the sign ± is determined by an algorithm, similar to the one used in the traditional Floer theory (see [13]). This algorithm shows, in particular, that the operations [·, …, ·]_{s} are skew-symmetric: a transposition of any two elements γ_{1}, γ′_{2} in the bracket changes the sign by (−1)^{\overline{γ_{1}}\overline{γ_{2}}}. It is important to point out that compactness theorems for holomorphic curves (see [18, 20, 11]) garantee that the operations [ · , …, ·]_{s} take values in polynomial functions (and not in formal power series).

The operation [ · , …, ·]_{s} which was just defined on the generators of Θ admits a unique extension to a skewsymmetric multilinear operation \underbrace{Θ^{H_{2}} ⊗ …, ⊗Θ^{H_{2}}}_{s} →

Θ^{H_{2}} which satisfies the Leibnitz rule:

(−1)^{t}[θ_{1}, …, θ_{l}θ′_{l}, …, θ_{s}]_{s} = θ_{l}[θ_{1}, …, θ′_{l}, …, θ_{s}]_{s} + (−1)\overline{θ_{l}}θ′_{l}[θ_{1}, …, θ_{l}, …, θ_{s}]_{s},

where t = ∑_{1}^{l−1} \overline{θ_{i}}. Let us now take a closer look to the operation [θ]_{1} which will

also be denoted by dθ, and called the differential of θ. Notice that it decreases the grading by 1, i.e. \overline{dθ} = \overline{θ} − 1 for any monomial θ ∈ Θ.

Theorem 3.4 1. d^{2} = 0.

2. Any homotopy α_{t}, t ∈ [0, 1], of contact forms, together with a compatible homotopy J_{t} of almost complex structures, induces a quasi-isomorphism Φ_{{a_{t},J_{t}}} of the corresponding algebras. In particular, the graded contact homology algebra HΘ^{H_{2}} = Ker d/Coker d is an invariant of the contact manifold (M, ξ).

Sometimes it is more convenient to consider the reduced contact homology algebra H̃Θ^{H_{2}}(M, ξ) of a closed contact manifold (M, ξ), which is defined similarly to HΘ^{H_{2}}, except that instead of contact forms for ξ on the whole M, we use contact forms on the punctured manifold M \ x, x ∈ M, which are isomorphic at infinity

to the standard contact form dz − ∑_{i=1}^{n−1} y_{i}dx_{i} on ℝ^{2n−1}.

The contact homology algebras HΘ^{H_{2}} and/or H̃Θ^{H_{2}} can be explicitely computed in several interesting examples. Let us formulate here some of these results.

Theorem 3.5 1. H̃Θ(S^{2n−1}, ξ_{0}) = ℂ; HΘ(S^{2n−1}, ξ_{0}) is a graded polynomial algebra of generators γ_{1}, γ_{2}, …, of degrees \overline{γ_{i}} = 2(n + i − 1), i = 0, ….

2. If ξ is an overtwisted contact structure on a 3-manifold V then HΘ(V, ξ) = 0.

3. For any Stein fillable contact manifold (V, ξ) we have HΘ(V, ξ) ≠ 0.

4. Suppose that a contact manifold (V, ξ) of dimension 2n−1 with H_{1}(V) = 0 has

a subcritical Stein filling W. Then H̃Θ^{H_{2}}(V, ξ) is a group algebra of H^{2}(V) over a free graded commutative algebra with generators γ_{ikl}, k = 1, …, n − 1, l = 1, …, dimH_{k}(W; ℝ), i = 0, …, of degree \overline{γ_{ikl}} = 2(n + i − 1) − k.

5. Let ξ_{1} be a non-standard contact structure on S^{2n−1}, n > 2, which is provided by Theorem 3.1 above, and the contact manifold (S^{2n−1}, ξ_{k}) be the connected sum of k copies of (S^{2n−1}, ξ_{1}). Then the contact homology algebras HΘ(S^{2n−1}, ξ_{k}) are pairwise non-isomorphic for all k, and in particular S^{2n−1} has infinitely many distinct contact structures in the standard homotopy class.

We thank Yu. Chekanov who pointed out to us the property 3.5.3. The computations in 3.5.4 were done by M.-L. Yau, and the result in 3.5.5 is due to I. Ustilovsky.

The case H_{1}(V) ≠ 0. For a general contact manifold V with H_{1}(V) ≠ 0 one may first construct a similar contact homology algebra HΘ_{contr}^{H_{2}} generated by the subset P_{a}^{contr} ⊂ P_{a} of contractible periodic orbits, and then for each free loop homotopy class Γ consider a module Θ_{Γ} over the algebra Θ_{contr}^{H_{2}}, generated by elements of P from the homotopy class Γ. The differential d : Θ_{Γ} → Θ_{Γ} on this module is defined, as above, by counting components of 1-dimensional moduli spaces M(γ; δ_{1}, …, δ_{r}) with an extra condition that γ and δ_{1} belong to the class Γ, while all the other trajectories δ_{2}, …, δ_{s} are from P^{contr}. Then we also have d^{2} = 0, and thus the homology HΘ_{Γ}^{H_{2}}, which is a module over the contact homology algebra HΘ_{contr}^{H_{2}} is another invariant of the contact manifold V. For a generic α HΘ_{Γ}^{H_{2}} is a finite dimensional module over HΘ_{contr}^{H_{2}}, and thus can be effectively computed, especially when the contact homology algebra HΘ_{contr}^{H_{2}} is isomorphic to ℂ (or to the group algebra of ℂ[H_{2}(V)]). For instance, let ξ_{1} be the standard contact structure on the 3-torus V = T^{3} = T^{2} × S^{1} viewed as the unit cotangent bundle of T^{2}. For k = 2, …, we denote by ξ_{k} the pull-back of ξ_{1} under the k-sheeted covering T^{3} → T^{3} which unwinds the fiber S^{1}. Then

Theorem 3.6 HΘ_{contr}^{H_{2}}(ξ_{k}) = ℂ[H_{2}(T^{3})]; dim_{ℂ[H_{2}(T^{3})]}(HΘ_{Γ}^{H_{2}}(ξ_{k})) = 2k for any horizontal 1-dimensional homology class, i.e. a class from H_{1}(T^{2} × point) ⊂ T^{3}, and HΘ_{Γ}^{H_{2}}(ξ_{k})) = 0 for all other classes Γ ∈ H_{1}(T^{3}).

As a corollory of 3.6 we get a theorem of E. Giroux and Y. Kanda (see [16, 21]) which states that the contact structures ξ_{k}, k = 1, …, are pairwise nonisomorphic. It seems likely that the algebra HΘ_{contr}^{H_{2}}(V, ξ) is trivial (i.e. isomorphic to the group algebra of H_{2}(V) over ℂ) for any strongly tight contact manifold (V, ξ), i.e. a contact 3-manifold which is covered by ℝ^{3} with a tight, and hence standard contact structure.

Hamiltonian formalism. It turns out that the the operations [ · , …, ·]_{s} for s > 1 can be viewed as certain cohomological operations on the contact homology algebra. For instance, we have

Theorem 3.7 The operation [ · , ·]_{2} is a Poisson bracket on Π(HΘ), where Π is the operator of changing the parity. The quasi-isomorphism Φ_{{a_{t},J_{t}}} from 3.4.2, induced by a deformation of contact forms and almost complex structures, preserves the Poisson bracket.

Other operations [ · , …, ·]_{s}, s > 2, define secondary cohomological operations on the contact homology algebra HΘ^{H_{2}}, which all fit into a structure of a L_{∞}, or rather a P_{∞}-algebra on Θ^{H_{2}}. However, the following Hamiltonian formalism provides a better algebraic framework for all these operations.

Let us associate with each periodic trajectory γ ∈ P = P_{α} two variables, p_{γ} and q_{γ} of the same degree \overline{p_{γ}} = \overline{q_{γ}}. Let T Θ be the free graded (super-)commutative algebra over ℂ with the unit generated by variables p_{γ}, q_{γ} associated to each periodic orbit γ ∈ P, and completed with respect to variables p_{γ}, γ ∈ P. This means that the elements of T Θ are formal power series in p-variables with coefficients which are polynomial of q-variables. We will also consider the group algebra T Θ^{H_{2}} of the group H_{2}(V) with coefficients in T Θ. Informally, if one thinks about the algebra Θ as the algebra of polynomial functions on an infinite-dimensional (super-)space L with coordinates q_{γ}, γ ∈ P, then T Θ is the algebra of functions on the cotangent bundle T^{∗}L of L. This infinite-dimensional cotangent bundle is endowed with an even symplectic form ∑_{γ∈P} dp_{γ} ∧ dq_{γ}, which defines, in its turn,

Poisson brackets on algebras T Θ and T Θ^{H_{2}}.

Next we construct a Hamiltonian function H ∈ T Θ^{H_{2}} which will encode all the information about the brackets [ ·, ⋯, ·]_{k} introduced above. We set

H(p, q) = ∑[γ_{1}, …, γ_{s}]_{s}p_{γ_{1}} ⋯ p_{γ_{s}},

where the sum is taken over the set of all monomials in variables p_{γ}, γ ∈ P, and where we assume that the brackets [γ_{1}, …, γ_{s}]_{s} ∈ Θ^{H_{2}} are expressed in terms of the variables q_{γ}, γ ∈ P. Theorems 3.8 and 3.9 below generalize Theorem 3.4 and formalize properties of all considered above operations.

Theorem 3.8 {H, H} = 0.^{6}

Consider a Poisson subalgebra ZΘ^{H_{2}} = {f ∈ T Θ^{H_{2}}; {f, H} = 0} ⊂ T Θ^{H_{2}} and its ideal BΘ^{H_{2}}, generated by functions of the form {g, H}, g ∈ T Θ^{H_{2}}. Then P Θ^{H_{2}} = ZΘ^{H_{2}}/BΘ^{H_{2}} also carries a Poisson structure.

Theorem 3.9 Any homotopy α_{t}, t ∈ [0, 1], of contact forms, together with a compatible homotopy J_{t} of almost complex structures, induces an isomorphism Ψ_{{α_{t},J_{t}}} : P Θ^{H_{2}}(V, α_{0}, J_{0}) → P Θ^{H_{2}}(V, α_{1}, J_{1}) of Poisson algebras.

These results are only the first steps of a bigger story. For instance, a directed symplectic cobordism between two contact manifolds generate a Lagrangian correspondence between the corresponding Poisson algebras, and the composition of directed cobordisms generates the composition of Lagrangian correspondences. We hope that this would provide tools for effective computations of rational Gromov-Witten invariants of symplectic manifolds by splitting them into compositions of elementary directed symplectic cobordisms. The larger picture also incorporates moduli spaces of holomorphic curves of higher genus, as well as higher-dimensional spaces of holomorphic curves.

Invariants of Legendrian submanifolds. Let us briefly mention here a relative analog of the contact homology theory, which provides invariants of pairs (V, L) where V = (V, ξ) is a contact manifold, and L its Legendrian submanifold. For the case when (V, ξ) is the standard contact (ℝ^{2n−1}, ξ_{0}) this theory produces invariants of immersed Lagrangian submanifolds in ℝ^{2n−2} up to contact isotopy (see [10]), i.e up to regular Lagrangian homotopy in ℝ^{2n−2}, which lifts to a Legendrian isotopy in ℝ^{2n−1}. When n = 2 all the involved holomorphic curves can be explicitely seen from the combinatorics of the corresponding (Lagrangian) immersion of the curve L into ℝ^{2}, and thus the theory may be developed via pure combinatorial means. The first part of this combinatorial theory, parallel to the theory of the differential d in the absolute case, was independently done by Yu. Chekanov (see [3]). However, even for n = 2 a (non-commutative) analog of the described above Hamiltonian formalism allows us to define many other invariants of Legendrian curves, which can also be computed and studied by pure combinatorial means.

6One should remember that in the super-commutative setting the bracket of a function with itself does not vanish automatically.

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References

[1] V. I. Arnold, On a characteristic class entering in quantization conditions, Funct. Anal. and Applic., 1(1967), 1–14.

[2] D. Bennequin, Entrelacements et équations de Pfaff, Astérisque, 106– 107(1983).

[3] Yu. Chekanov, Differential algebra of a Legendrian link. Preprint 1997.

[4] Y. Eliashberg, On symplectic manifolds which some contact properties, J. of Diff. Geom., 33(1991), 233–238.

[5] Y. Eliashberg, New invariants of symplectic and contact manifolds, J. of AMS, 4(1991), 513–520.

[6] Y. Eliashberg, Classification of contact structures on ℝ^{3}, Int. Math. Res. Notices, 3(1993), 87–91.

[7] Y. Eliashberg, Classification of overtwisted contact structures, Invent. Math., 98(1989), 623–637.

[8] Y. Eliashberg, Contact 3-manifolds twenty years after J. Martinet’s work, Annales de l’Inst. Fourier, 42(1992), 165–192.

[9] Y. Eliashberg, Unique holomorphically fillable contact structure on the 3torusInternat. Math. Res. Notices 1996, no. 2, 77–82.

[10] Y. Eliashberg and M. Gromov, Lagrangian intersections theory. Finitedimensional approach, Arnold’s volume, AMS, Providence 1998. 1996.

[11] Y. Eliashberg, H. Hofer and S. Salamon, Lagrangian intersections in contact geometry, Geom. and Funct. Anal., 5(1995), 244–269.

[12] Y. M. Eliashberg and W. P. Thurston, Confoliations, University Lecture series, 13(1998), AMS, Providence.

[13] A. Floer, H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z., 212(1993), 13–38.

[14] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariants, preprint 1996.

[15] H. Geiges, Applications of contact surgery, Topology, 36(1997), 1193–1220.

[16] E. Giroux, Une structure de contact, même tendue est plus ou moins tordue, Ann. Scient. Ec. Norm. Sup., 27(1994), 697–705.

[17] J.W. Gray, Some global properties of contact structures, Annals of Math., 69(1959), 421–450.

[18] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent.

Math., 82(1985), 307–347.

Documenta Mathematica · Extra Volume ICM 1998 · II · 327–338

[19] M. Gromov, Partial Differential Relations, Springer-Verlag, 1986.

[20] H. Hofer, Pseudo-holomorphic curves and Weinstein conjecture in dimension three, Invent. Math., 114(1993), 515–563.

[21] Y. Kanda, The classification of tight contact structures on the 3-torus, preprint 1995.

[22] M. Kontsevich, Deformation quantization of Poisson manifolds, I, preprint 1997.

[23] G. Liu and G. Tian, Floer homology and Arnold conjecture, preprint 1997.

[24] R. Lutz, Structures de contact sur les fibrés principaux en cercles de dimension 3 Ann. Inst. Fourier, XXVII, 3(1977), 1–15.

[25] J. Martinet, Formes de contact sur les variétés de dimension 3, Lecture Notes in Math., 209(1971), 142–163.

[26] J. Robbin, D. Salamon,The Maslov index for paths, Topology, 32(1993), 827– 844.

Yakov Eliashberg

Department of Mathematics Stanford University Stanford, CA 94305-2125 USA eliash@math.stanford.edu