Differential Geometry and its Applications 9 (1998) 59-88 North-Holland

Symplectic topology in the nineties*

Yasha Eliashberg Stclntbrd CJniver.sity Sturford CA 94305, USA

Communicated by M. Gotay Received 18 January I998 Revised 5 May 1998

Ahstruc-r: This is a survey of some selected topics in symplectic topology. In particular, we discuss low

dimensional symplectic and contact topology, applications of generating functions, Donaldson’s theory oi approximately complex manifolds and some other recent developments in the field.

Kcwwdx Symplectic and contact structure, Lagrangian submanifold. pseudo-holomorphic curves.

M.S c~lus.s~fic.utinn: S3C IS. 58FO5.

Twenty years ago Symplectic Topology did not exist. When ten years ago I wrote a short

survey [ 51, it was possible at least to indicate the main directions of the new field. The develop-

ment of Symplectic Topology during the last decade can only be characterized as an explosion.

Today it is a melting pot of ideas from Physics, Topology, Algebraic and Differential Geometry,

Analysis. etc.

In the current survey we discuss some of the new developments without any attempt to be

complete. The choice of subjects more reflects the author’s current interests rather than the

objective state of the art in the field.

1 want to thank P. Biran for his help in the preparation of this survey. I am also grateful to the

referee and the editor for helpful critical remarks.

1. Symplectic basics

I. I. Symplectic and contact manifolds

A symplectic structure on a 2k-manifold W is a closed non-degenerate differential 2-form o.

The non-degeneracy condition means that the formula I(,,( t ) = w (r, ), T E T(W), defines

an isomorphism Z, : 7’(W) + T*(W) between the tangent and cotangent bundles of the

manifold W. For a 2-dimensional W a symplectic form is just an area form of the surface.

According to Darboux’ theorem any symplectic form is locally equivalent to the canonical

,form COO = Cl; dx; A dy,. Thus, equivalently a symplectic manifold can be characterized by

existence of local Darboux charts glued together by symplectomorphisms, i.e., diffeomorphisms

which preserve the canonical form. A 1 -form CY on a (2k - I)-dimensional manifold V is called contact if the restriction of da

to the (2k - 2)-dimensional tangent distribution 6 = {a = 0) is non-degenerate (and hence

* Partially supported by the NSF grant DMS-9626430.

09262245/98/$19.00 @ 199X-Elsevier Science B.V. All rights reserved PI1 SO926-2245(98)0003 8-7

60 I! Eliashberg

symplectic). Equivalently, we can say that a l-form Q is contact if u A (d~)~-’ does not vanish

on V. A codimension 1 tangent distribution 6 on V is called a contact structure if it can be locally

(and in the co-orientable case globally) defined by the Pfaffian equation c! = 0 for some choice

of a contact form a. The pair (V, e) in this case is called a contact manifold. Note that according

to Frobenius’ theorem the contact condition is a condition of maximal non-integrability of the

tangent hyperplane field c. In particular, all integral submanifolds of c have dimension 6 k - 1.

On the other hand, (k - 1)-dimensional integral submanilods, called Legendrian, always exist

in abundance. Any non-coorientable contact structure can be canonically double-covered by a

coorientable one. If a contact form o. is fixed then one can associate with it the Reeb vector

jield R,, which is transversal to the contact structure c = {(Y = O}. The field R, is uniquely

determined by the equations

R,Ada = 0; cx(Ra) = 1.

A symplectic structure w on W defines a volume form mk, and hence an orientation of W. In

particular, if W is closed then the cohomology class [w] E H2( W; IK) represented by the closed

form w satisfies the inequality [wlk # 0.

If a contact structure < is defined by a l-form a, then the conformal class of the symplectic

structure dcxlc depends only on e (because d(fa) 1~ = fdcxlc for a function V + Iw); we

denote it by CS(,$). For an even integer k the contact structure defines an orientation of the

(2k - 1)-dimensional manifold V; if k is odd, it defines an orientation of e.

An important example of a symplectic manifold is provided by the cotagent bundle T*(M)

of any smooth manifold M. The symplectic form w on T*(M) is the differential of the famous

l-form p dq. Alternatively the symplectic structure on T*(M) can be described as follows.

If M = IKk then IR2k = T*(IEk) is endowed with the canonical symplectic structure wg =

d(p dq) = c’; dpi A dqi, where the coordinates q = (ql, . . . , qk) and p = (~1, . . . , pk) are

chosen in such a way that the projection T*(IKk) + IWk is given by (p, q) H q. Let us observe

that any diffeomorphism f : JRk + Rk lifts to a symplectomorphism f* : T*(IWk) + T*(EXk)

by the formula

A(P, q) = (f(q), (df*)-l(p)).

Thus a coordinate atlas M = l_lj Uj on M lifts to a symplectic atlas T*(M) = Uj T*(Uj) with

gluing symplectomorphisms lifted by the above formula.

The standard contact structure to on IR 2k-1 is defined by the contact l-form dz - Et-’ pi dqi

in the coordinates (41, . . . , q&l, ~1, . . . , pk_1, z). More generally, the space J’(M) = T*(M) x Iw of l-jets of functions on A4 has the canonical contact structure defined by the

contact form dz - p dq on T*(M) x Iw, where the coordinate z corresponds to the second

factor, and where we identify the form p dq on T*(M) with its pull-back on J’(M).

Another important example of a contact manifold is provided by the space ofcontact eEements of a smooth manifold M, or in other words, by the projectivized cotangent bundle P T * (M). A

point of P T*(M) is a tangent hyperplane to M which can be identified with a line in T*(M).

The canonical 1 -form p dq does not descend to P T* (M), but its kernel does and this defines a canonical contact structure on P T*(M). This contact structure is not co-orientable. The double

cover of PT*(M), which carries a co-orientable contact structure, is the associated spherical

Symplectic topology in the ninetie. hl

bundle ST*(M) which can be viewed as the space of co-oriented tangent hyperplanes (elements).

Similar to the symplectic case, contact manifolds have no local geometry: any (2k - l)-

dimensional contact manifold is locally isomorphic to the standard contact R2k-‘; moreover,

any contact form is locally isomorphic to the standard contact form dz - Cl;-’ pi dqi.

Complex geometry serves as a rich source of examples of symplectic and contact manifolds.

An imaginary part of a Kahler metric is a symplectic form. In particular, the standard symplectic

structure on the complex projective space CP”, which in homogeneous coordinates (za : . . : z,~)

equals to

w = $3 log2 lz& k=O

is the imaginary part of the Fubini-Study metric. Here the coefficient for w is chosen to satisfy

the condition so,, =ePn w = I.

A strictly pseudo-convex hypersurface in a complex manifold carries a canonical contact

structure defined by the maximal complex subbundle of its tangent bundle. See Section 2 below

for more details.

Contact and symplectic structures on closed manifolds satisfy the following stability property,

which is due to J. Gray [29] in the contact case, and to J. Moser 1361 in the symplectic one.

Theorem 1.1. Let M be a closed manifold, & a.family of contact structures on M, or w, a

,family of symplectic structures on M with [o,] = const, t E [0, 11. Then there exists an isotopy

fr : M + M, such that dft(.&) = &, or fF(wo) = wy, t E [0, 11.

1.2. Lagrangian and Legendrian submanifolds

Lagrangian submanifolds of symplectic manifolds and Legendrian submanifolds of contact

manifolds play the central role in Symplectic Topology.

A k-dimensional submanifold L c W of a 2k-dimensional syrnplectic manifold (W, w) is

called Lagrangian if the form W]L vanishes.

A (k - I)-dimensional submanifold A c W of a (2k - 1)-dimensional contact manifold V is

called Legendrian if A is tangent to 6. Equivalently, A is Legendrian if for each point q E A the

tangent space T,)(A) is a Lagrangian subspace of e,, with respect to a symplectic structure from

C’S(c). Similarly, one can define Lagrangian and Legendrian embeddings and immersions.

.4 section s : M + T*(M) is a Lagrangian embedding iff s is a closed l-form. In fact, any Lagrangian submanifold L of T*(M) can be viewed as a multi-valued closed form, or

better to say that the closed form p dq ]L. is defined on the submanifold L itself rather than on its

projection. The Lagrangian submanifold L c T*(M) is called exact if the closed l-form p dq 1 L

is exact. A section o : M -+ J’(M) is a Legendrian embedding iff c is the l-jet extension

j ’ ( f) of a function f : M --+ R.

Similarly to the case of Lagrangian submanifolds of the cotangent bundle, a general Legen-

drian submanifold of J’ (M) corresponds to a graph (“wave front”) of a multivalued function. A projection J ’ (M) = T*(M) x R + T*(M) sends Legendrian submanifolds of J ’ (M)

onto (immersed) exact Lagrangian submanifold of T*(M). Conversely, any exact Lagrangian

62 E Eliashberg

submanifolds of T*(M) lifts, uniquely up to a translation along the R-factor, to a Legendrian

submanifold of J’(M). The Legendrian submanifold ,&c, = j’(f)(M) c J’(M), and its ex-

act Lagrangian projection Lf = df(M) c T*(M) are called graphical Legendrian or exact

Lagrangian submanifolds.

1.3. Hamiltonian functions, vectorjelds and difeomorphisms

A vector field X on a symplectic manifold (M, w) is called symplectic if the Lie derivative

kcxo vanishes, which is equivalent to the equation d(X 1 o) = 0. If the form X 1 w is exact,

i.e., X J w = dH, then the vector field X = XN is called Hamiltonian, and the function H

is called the Hamiltonianfunction for the vector field X. If M is non-compact then we will

always assume that X and H have compact supports. This condition defines H uniquely. If M

is compact then H is defined up to adding a constant. A time-dependent Hamiltonian function

HI : A4 + IR, t E [0, I], defines a symplectic isotopy qPt : M + M, t E [0, 11. Namely, pal is

determined by the differential equation

&t(x) ___ = Xff,(Ipl(X))>

dt t E [O, 11, x E M,

with the initial condition qoa(x) = x, x E M. The isotopy po, is called a Hamiltonian isotopy with

the time-dependent Hamiltonian function H,, t E [0, 11. A symplectomorphism p : M + M

is called Hamiltonian if it is the time 1 map of a Hamiltonian isotopy. The group Ham of

Hamiltonian diffeomorphisms of (M, o) is a normal subgroup of the identity component Diff,

of the group of symplectomorphisms of (M, w), and we have Ham = [Diff,, Diff,], see [25].

Hamiltonian diffeomorphisms can be recognized among all symplectomorphisms with the

help of the Flux, or Calabi homomorphism (see [28] and also [25,13]), which is defined as

follows. Let l%& be the universal cover of the group Diff,. Thus an element of KffW is a

homotopy class of a path par E Diff,, t E [0, I], where the homotopy fixes the ends of the path.

Given a path CJJ* E Diff,, t E [0, 11, we denote by X, the symplectic vector field

and set

Flux({~o,l) = j,x, JOI dt, 0

where [X, J w] E H’ (M; IR) is the cohomology class of the closed l-form X, J w,

t E [0, 11. It is straightforward to check that Flux({~~}) depends only on the homotopy class of

a path connecting Id and ~1, and thus Flux is a homomorphism DxW -+ H’(M; IR). A sym-

plectomorphism p E Diff, is Hamiltonian if and only if Flux(@) = 0 for some lift 40 E l%fffW

of 40 E Diff,.

A. Banyaga [25] proved that if the manifold A4 is closed then the group Ham(M, w) is simple. However, in the non-compact case this is no longer true. Namely, Ham has the commutator

Sytnplectic topology in the nineties 63

subgroup ]Ham, Ham] as a proper normal subgroup which is equal to the kernel of the Culubi

homomorphism, which can be defined as follows.

Let f E Ham be the time 1 map of a Hamiltonian isotopy generated by a time-dependent

Hamiltonian function H, Then Cal(f) = J,,, H f con dt In the general case this integral depends

on the homotopy class of the path chosen to connect f with the identiity, and thus Cal is a

homomorphism H% + IR, where H% is the universal cover of the group Ham. However. in

the case when the symplectic structure w is exact, this homomorphism descends to the group

Ham itself (see [28] and [25] for the details).

1.4. Symplectic and &most complex structures

Let w be a symplectic bilinear form on a real vector space V. A complex structure J : V -+ V.

J’ = -Id, on V is called compatible with w if the form w is invariant with respect to J and

o (A’, J X) > 0 for any non-zero vector X E V. In other words. this can be expressed by saying

that

o(X, JY) - iw(X, Y), x. Y E v,

is a positive definite Hermitian form on the complex vector space (V. J). If just a weaker

condition

w(X, JX) > 0 for any tangent vector X # 0

is satisfied, than according to Gromov’s definition (see [48]) the complex structure J is tamed

by UJ. The group Sp( V, w) of symplectic automorphisms acts on the space a of compatible

complex structures with the subgroup U(V, J) of unitary transformations as the stabilizer

subgoup. Thus g can be identified with the homogeneous space Sp(V, w)/U(V, J) which is

contractible, as can be seen, for instance, from the polar decomposition. The space of complex

structures tamed by w has 2 as its deformation retract, and hence is also contractible.

Coming to a non-linear situation let (W, w) be a symplectic manifold. We denote by J( W. w)

the space of all almost complex structures J : T(W) + T(W) compatible with 01~~ (w, on

every tangent space 7” (W), x E W. The space g( W, (0) is non-empty and contractible, as it is

the space of sections of a fibration with contractible fibers. It is also important to notice that the

space of symplectic structures compatible with a given almost complex structure is a convex

subset of a vector space, and therefore also contractible. This space is non-empty if W is open

(see 171). In the case of a closed manifold W existence of an almost complex structure is far

from being sufficient for existence of a symplectic structure on W (see Problem 2 in Section 7).

A generic almost complex manifold has no complex submanifolds of (complex) dimension

> 1 However, it was observed already by A. Nijenhuis and W. Wolf (see [222]) that locally any

almost complex manifold has as many holomorphic curves as one has in the integrable case.

M. Gromov (see [48]) was the first who understood that in the presence of a taming symplectic

form one can develop a global theory of holomorphic curves, similar to the case of (inte-

grable) Kahler manifolds. His introduction of tamed (or compatible) almost complex structures into Symplectic Geometry revolutionized this area. The theory of pseudo-holomorphic curves,

as introduced by Gromov, forms the foundation for most recent developments in Symplectic

64 I! Eliashberg

Topology. In this survey we will omit the prefix “pseudo” and use the term “holomorphic” for

curves in almost complex manifolds.

1.5. Symplectic rigidity

One of the basic facts, which Gromov proved [48] using his theory of holomorphic curves

in symplectic manifolds is the following

Theorem 1.2. The groups of symplectic and contact difleomorphisms are Co-closed in the

groups of all diffeomorphisms.

This result, which establishes the existence of symplectic topology, was proved earlier by the

author (see [47]) using a combinatorial theorem about the structure of wave fronts, but the proof

of this combinatorial theorem was not published. Symplectic rigidity could also be deduced

from Conley-Zehnder’s work on Arnold’s conjecture for T2” (see [45]).

A related fact, also proven in Gromov’s seminal paper [48], is the existence of a first specif-

ically symplectic invariant, which is now called Gromov’s width. This invariant can be defined

as follows. Let (M, o) be a symplectic manifold (for instance, a domain in the standard sym-

plectic IR2n). Fix a point p E M. Given an almost complex structure J on M tamed by w, and

a J-holomorphic curve C c A4 which is a closed subset of M and passes through the point p,

set

A(C,J,p)=/~

c

and define Gromov’s width as

w(M, w) = supinf A(C, J, p) . J c

Notice that w (M, w) is independent of the choice of the point p because the symplectomorphism

group of (M, w) acts transitively on M.

The following theorem (see [48]) summarizes the properties of Gromov’s width.

Theorem 1.3. 1. For any symplectic (M, w) we have 0 < w(M, o).

2. w(M, k0) = A’w(M, 0) for any h > 0.

3. w(D2”(r)) = w(D2(r) xJR2n-2) = nr2. Here D2”(r) is the ball of radius r in R2”, which

is endowed with the standard symplectic structure.

As a corollary we get the following famous Gromov non-squeezing theorem

Corollary 1.4. Zfthere exists a symplectic embedding D2”(1) + D2(r) x iR2n-2 then r > 1.

The theory of symplectic invariants pioneered by Gromov was later developed by Ekeland-

Hofer [ 1621 and Hofer-Zehnder [ 1691, and culminated in Floer-Hofer symplectic homology the-

ory, which they developed jointly with K. Cieliebak and K. Wysocki (see [165,160,167,161]). C. Viterbo (see [196]) found an alternative finite-dimensional approach to the theory of sym-

plectic invariants (see also Section 6 below).

Symplectic topology in dze nineties 65

2. Contact 3-manifolds

A contact structure on a 3-dimensional manifold is easy to visualize. However, contact

geometry in dimension 3 is a great source of interesting and difficult symplectic geometric

problems.

To fix the stage we restrict our discussion to co-orientable positive contact structures 6 =

{CX = 0}, where the positivity means that the contact orientation defined by the form a A da

coincides with an a priori given orientation of the manifold.

It is known since the work of J. Martinet ([35]) and R. Lutz ([33]) that any orientable contact

manifold admits a contact structure in every homotopy class of tangent plane fields. However, as

was first discovered by D. Bennequin ([44]), even on S3 there exists a homotopy class of plane

fields which can be represented by non-isomorphic contact structures. The phenomenon which

causes this non-uniqueness is called overtwisting and was studied in [54]. It turned out that it is

useful to distinguish two complementary classes of contact structures. A contact structure t on

a 3-manifold M is called over-twisted if there exists an embedded 2-disc D c M bounded by a

Legendrian curve, and which is transversal to < along a D. A non-over-twisted contact structure

is called tight. It was shown in [54] that classification of overtwisted contact structures up to

isotopy coincides with their homotopical classification as plane fields. Therefore, it seems that

overtwisted contact structures do not exhibit enough rigidity to make them useful for serious

geometric applications.

Tight contact structures are much more rigid. They also seem to be much more useful (see,

for instance, proof of Cer-f’s theorem “r4 = 0” in [55]), and thus more difficult to understand.

2.1’. Clussification results

The following theorem gives a nearly exhaustive description of classification results for tight

contact structures. known at this moment.

Theorem 2.1. 1. The following manifolds:

s3, IRP3, s* x s’,

and each of these manifolds minus a finite number of points, have unique, up to isotopy, tight

contact structures (see [55]).

2.Let$1 bethecanonicaEcontactstructureonT3 = ST*(T2) = T2xS’,and&,,n = 2,. . .

be thepull-backofthe structure e1 by the covering map Id x rr,, : T3 = T2 x SE --+ T2 x S1 = T’,

where n,(u) = nu, u E R/Z = S’. The structures &,, n = 1, . . . , are tight, pair-wise non-

isomorphic and comprise the complete list of tight contact structures on T3, up to isomorphism

(see [64,67]).

3. A connected sum Ml#M2 of two tight contact manifolds (M, 61) and (M, &) admits a

unique tight contact structure 6 = .$I#& compatible with the contact structures of the summands.

Conversely, if a manifold M splits as a connected sum M = M1#M2 then any contact structure

splits uniquely up to isotopy into the connected sum < = ,$,#& (see [55]).

66 E Eliushberg

Recently, E. Giroux extended the classification result in Theorem 2.1.2 from T3 to other

toric fibrations over S’, and J. Etnyre (see [60]) classified tight contact structures on certain

Lens spaces. In particular, he proved that any Lens space L(p, s) admits only finitely many

non-isomorphic tight contact structures. Notice also that 2.1.2 classifies contact structures on

T3 only up to an isomorphism, i.e., a preserving contact structure diffeomorphism which need

not to be isotopic to the identity. The classification of contact structures on T3 up to isotopy is

also known (see [64] and [178]).

As Theorem 2.1.2 shows, the number of isomorphism classes of tight contact structures on

a given manifold need not be finite, even in the same homotopy class of plane fields. However,

this phenomenon could be related to the presence of incompressible tori: I do not know any

atoroidal manifold with infinitely many non-isomorphic contact structures. (A surface in a 3-

manifold is called incompressible if its fundamental group injects into the fundamental group

of the ambient manifold.) It is also likely that on any 3-manifold only finitely many homotopy

classes of tangent plane fields can be realized by tight contact structures. Although this is still

unknown, a result of P. Kronheimer and T. Mrowka (see [68]) establishes the finiteness of

homotopy classes of plane fields realizable by symplectically semifillable structures (see the

next section for the definition of semi-fillabillity).

Let us also mention here a recent result by V. Colin (see [53]) which states that Co-close tight

contact structures are isotopic.

2.2. Recognizing and constructing tight contact structures

It is not easy to verify whether a contact structure is tight. Even the tightness of the standard

contact structure on S3 is a highly non-trivial fact which was proven by D. Bennequin in 1982

(see [44]).

A contact 3-manifold (M, e) is called symplecticallyjillable if there exists a compact sym-

plectic manifold ( W, o) with boundary, such that

- aw=M; - the form uI~ does not vanish; - the contact orientation of (M, t) coincides with the orientation of M as the boundary of

the symplectically oriented manifold (W, 0).

The manifold (M, 6) is called symplectically semi-jillable if it is a connected component of a symplectically fillable contact manifold. The following theorem (see [48] and [59]) provides

an effective criterion for tightness.

Theorem 2.2. A symplectically semi-jillable contact structure is tight.

An important class of symplectically fillable manifolds consists of holomorphicallyJillable

ones. Any contact plane bundle t = (a = 0) admits a structure of a complex bundle, such that the complex structure J : $ + 6 is compatible with the symplectic form dcz/, (see Section 1.4

above). Moreover, in the 3-dimensional case this J, called a CR-structure, always can be chosen

integrable and extendable as a complex structure to W = M x (E, E) > M x 0 = M. We assume here that the splitting of W is chosen in such a way that the vector field JR,, where R, is the Reeb vector field determined by the l-form a, is an inward transversal to the boundary M x 0

Swnplectic topology in the nineties 67

of the domain W+ = M x (--E, 01. From the complex analytic point of view this means that

M x 0 is a strictl),pseudo-convex boundary of the domain W+ = M x (--F, 01. If the complex

manifold ( W, J) extends to a compact complex manifold W with boundary A4, then the contact

structure < is called holomorphically fillable. In this case the manifold W is actually Ktihler.

and hence symplectic, and the contact manifold (M, 6) serves as a contact boundary of the

symplectic manifold W. All the structures on closed 3-manifolds mentioned in Theorem 2.1

from Section 2. I, except the structures tn on T’ for y1 > I are holomorphically fillable. On

the torus T’ the standard structure ,$ is a unique, up to a contactomorphism, holomorphically

fillable contact structure (see [56]).

Theorem 2.3 below (see 12031) gives a complete, although not very constructive description

of all holomorphically fillable contact structures.

First notice that any embedded Legendrian circle S in a contact 3-manifold (M, < = (a, = 0))

admits a canonical framing. Namely, choose an orientaion of S by a tangent vector field r (the

final result of the construction will be independent of the choice of the orientation), and take the

framing (R,, Jr). It is independent, up to homotopy, of the choice of the contact form Q and a

compatible complex structure J : 6 + 6. It is shown in [203] (see also 12361) that the Morse

surgery along S with respect to a framing which differs from the canonical one by the rotation

by -2n, can be performed, in a unique way, in the category of holomorphically fillable contact manifolds.

Theorem 2.3. Any holomorphicallyJillable contact manifold can be obtainedfrom the standard

contact structure on the connected sum of k copies of S2 x S’ by a sequence of Legendrian

surqerirs.

Notice that any knot is isotopic to a Legendrian one. Moreover, if one gets a Legendrian

realization of a knot with certain framing then all framings which differ by a rotation by a negative

multiple of 2~ can also be realized. This observation shows that there are a lot of 3-manifolds

which carry holomorphically fillable contact structures. In [66] R. Gompf systematically studied

what kind of manifolds one can obtain by this construction. In particular, he proved

Theorem 2.4. a) Any oriented SeifertJiber space carries u holomorphicallyfillable, possibl>

negative contact structure.

h) An oriented Seifertjber space carries a positive holomorphically,fillable contact structure. exl ‘ept possibly the case when the base of thejbration is S2.

Gompf’s actual theorem provides more constructions of holomorphically fillable contact struc-

tures, and more detailed description of exceptional cases in 2.4a.

The theory of 2-dimensional foliations on 3-manifolds provides a rich source of constructions

of symplectically semifillable contact structures. In the discussion below we do not distinguish between codimension one foliations and integrable plane fields on 3-manifolds. Let us recall

that an important class of foliations on 3-manifolds is formed by tautfoliations. A foliation 6 is

called taut if there exists a closed transversal curve which intersects all leaves of the foliation.

Equivalently, taut foliations can be characterized by existence of a closed 2-form o such that

wji- nowhere vanishes.

68 YI Eliashberg

Taut foliations are contained in a larger class of Reebless foliations, i.e., foliations without

Reeb components. The following theorem is proved in [6].

Theorem 2.5. a) Any co-orientable foliation, except the foliation of the product S2 x S1 by the

horizontal 2-spheres, can be Co-approximated by contact structures, both positive and negative.

b) A contact structure, Co-close to a taut foliation is symplectically semi-jillable.

c) A contact structure, Co-close to a Reebless foliation is tight.

In [207] D. Gabai constructed a lot of taut foliations on 3-manifolds. In particular, he proved

that if a surface in an irreducible 3-manifold has minimal genus > 0 in its homology class then

it can be included as a leaf into a taut foliation.

It is possible that Theorems 2.3 and 2.5 provide enough tools to construct tight, or sym-

plectically (semi-)fillable contact structures on all irreducible orientable 3-manifolds. Notice,

however, that recently P. Lisca (see [7 11) proved that the Poincare homology 3-sphere P with one

of its orientations has no positive symplectically semi-fillable contact structure. It follows then

from Theorem 2.1.3 that P#(- P) has no symplectically semi-fillable contact structure at all.

3. Hofer geometry

One of the most remarkable manifestations of symplectic rigidity is the existence of a bi-

invariant metric on the group of Hamiltonian symplectomorphisms. The existence of a bi-

invariant metric is highly unusual for non-compact groups of transformations. For instance, the

group of volume preserving diffeomorphisms of a manifold of dimension > 3 does not admit

such a metric, as there exist volume preserving diffeomorphisms whose conjugacy classes

contain the identity in their Coo-closure.

This metric was first discovered by H. Hofer in his seminal paper (see [ 1061) which laid the

foundation of what is now called Hofer geometry.

Let Ham = Ham(M, o) be the group of (compactly supported) Hamiltonian symplecto-

morphisms of a symplectic manifold (M, w). Given a Hamiltonian isotopy q+, defined by a

time-dependent Hamiltonian function Hr : A4 -+ R, t E [0, 11, we set

II]qot]ll = ~pH&) - ixnfM-4,

where the sup and inf are taken over all (x, t) E A4 x [0, 11. We further set for q E Ham

where the inf is taken over all compactly supported Hamiltonian isotopies {q+} with vi = (0. It is

easy to check that ]I . II is a seminorm, and that p (f, g) = I I f g-’ I I is a bi-invariant pseudo-metric

on the group Ham. However, it is highly non-trivial to show that

Theorem 3.1. The semi-norm II . II is non-degenerate, i.e., defines a norm, which is called the

Hofer norm, on the group of Hamiltonian symplectomorphisms.

The non-triviality of the Hofer metric is a corollary of the following statement, called the energy-capacity inequality, which is very interesting by itself. Given a compact subset A c M

Symplectic topology in the nineties 69

(the manifold M is assumed to be without boundary) we define the displacement energy of A

as

e(A) = inf(]lcpl]; cp E Ham(M), q(A) f-’ A = VI}.

We also define a capacity of A as

c(A) = sup{nr2; there exists a symplectic embedding B’“(r) into IntA}

where B’“(r) is the ball of radius r in the standard symplectic IR?.

Theorem 3.2. For any compact set A c M one has the inequality

e(A) > it(A).

Theorems 3.1 and 3.2 were first established by H. Hofer (see [ 106]), and independently but

slightly later by C. Viterbo (see [ 1961) for the case when (M, w) is the standard symplectic XZn.

Notice that in this case the coefficient i can be dropped from the inequality in Theorem 3.2.

These results were later generalized by the efforts of many people to the case of a general

symplectic manifold, and in the final form were proven by F. Lalonde and D. McDuff in [ 1 lo].

It is not difficult (see, for instance, [ 1131) to check that

= i;,f s

(ma; H,(x) - $i; H,(x)) dt.

0

The latter equality shows that the Hofer metric p defines a length structure in the sense of

Gromov (see [212]) on the group Ham. In other words, one gets the Hofer metric starting with

the LX-type norm on the Lie algebraof the group Ham, which is just the space P(M) of smooth

functions on M. Next, one computes the length of a path by integrating the length of its velocity

vector, and finally defines the distance between two diffeomorphisms as the infimum of lengths

of connecting paths. A natural question is, what is the general class of norms on C”-functions

which produces via this construction bi-invariant metrics on the group of symplectomorphisms.

The bi-invariancy condition just means that the norm is invariant under symplectic changes of

coordinates. However, as it is shown in [ 1051, most norms generate only pseudo-metrics, usually

identically equal to 0, i.e., the Hofer metric is essentially unique. However, the problem of the

complete description of bi-invariant non-degenerate (Finsler) metrics on the group Ham is still

open.

Hofer geometry is an alive and active area. An interesting question studied by many mathe-

maticians concerns the diameter of the group Ham(M, w) in Hofer metric. Is diam(M, o) finite

or infinite? It is infinite in all known cases. In the case when (M, w) is non-compact it is easy

to see that diam(M, w) = co, because the Hofer norm ]]fll is bounded below by the absolute value of the Calabi invariant Cal(f) ( see 1.3 above) which obviously can take arbitrarily large values. For closed manifolds this is more subtle. F. Lalonde and D. McDuff (see [108]) proved

the infiniteness of the diameter for all surfaces of genus > 0, as well as for high-dimensional

tori. This result was extended by M. Schwarz [ 1751 to many symplectically aspherical manifolds

70 I! Eliashberg

(i.e., manifolds satisfying the condition o]~?(M) = 0). Recently L. Polterovich [ 1131 developed

new techniques for dealing with the spherical case and proved the infiniteness of the diameter

for a large class of manifolds, including S* and @P*.

Let me also mention here the results of M. Bialy and L. Polterovich ([ 156,157, lSS]) relating

Hofer geometry and Hamiltonian dynamics, and the results of I. Ustilovsky [ 1141, M. Bialy and

L. Polterovich [ 1041 and F. Lalonde and D. McDuff [ 1081 on the structure of geodesics in Hofer

geometry.

4. Donaldson’s theory of approximately complex submanifolds

Symplectic manifolds have a lot of symplectic submanifolds of codimension > 2. As was

shown by M. Gromov (see [30] and [7]) one has the following h-principle for symplectic

embeddings:

Theorem 4.1. Let (MO, WO) and (MI, 01) be two symplectic manifolds of dimension 2n and

2(n + k), k > 2, respectively. Suppose that an embedding f : A40 + MI satisJies the following

two conditions: _ the cohomology classes of the forms 00 and f *WI coincide;

- the differential df : T (MO) -+ T (MI) is homotopic through injective homomorphisms to

a symplectic homomorphism.

Then f is isotopic to a symplectic embedding.

This h-principle fails completely for symplectic embeddings in codimension two. However,

S. Donaldson proved recently a surprising theorem, analogous to the Kodaira embedding theo-

rem in Klhler geometry. In particular, Donaldson’s result provides us with an effective tool for

the construction of codimension two symplectic submanifolds, which are analogous to hyper-

plane sections of complex projective varieties.

Let (M, w) be a closed symplectic manifold and suppose that the cohomology class [o]

of its symplectic form is integral. Let Lk + M be a complex line bundle whose first Chern

class is equal to k[w], k E Z. The bundle Lk admits an Hermitian metric and a compatible

connection c, which is also compatible with the complex structure on Lk, and whose curvature

form equals kw. The connection c, together with the complex structure of the bundle and an almost complex structure on M, compatible with o, defines an almost complex structure J on

the total space Lk of the line bundle. This almost complex structure is non-integrable in general,

and in particular, one cannot hope to have holomorphic sections M -+ Lk. However, Donaldson

proved that one can still construct approximately holomorphic sections, and the larger k is, the

better approximation can be obtained. Approximate holomorphicity of a section s : A4 --+ Lk

means that the differential df : T(M) --+ T (Lk) does not deviate much from a complex linear

homomorphism, or more precisely, that ) 3s 1 < 1 i3s 1. If one can make s sufficiently transversal

to the O-section, so that the angle between the section and the O-section is much larger than the

deviation of s from a holomorphic section, then s -’ (0) will be an approximately holomorphic,

and hence symplectic submanifold of codimension 2. Donaldson realized this approach using

Yomdin’s refinement of Sard’s theorem. Here is Donaldson’s result (see [ 1381).

Symplectic topology in the nineties 71

Theorem 4.2. Let (M, w) and Lk be as above. Then for a st@ciently large k there e.rists

an approximately holomorphic section s : M + Lk suflcientlv transversal to the O-section.

In particular, the homology class from H2,,_2(M, R), dual to k[w] can be represented by a

symplectic submanifold. Moreover, all symplectic submanifolds obtained by) this construction

are Humiltonian isotopic. The real-valued function -log]s 1’ on &? = M \ s- ’ (0) has critical

points of index < n, and in particular z is homotopy equivalent to an n-dimensional cell

complex.

Very recently S. Donaldson ([ 1391) announced further developments in his theory of approxi-

mately holomorphic sections. In particular, one of his new results asserts existence ofsymplectic

singular fibrations, similar to Lefschetz pencils in Algebraic Geometry.

Here is one of Donaldson’s new results in this direction.

Theorem 4.3. Let (M. u) be a closed symplectic 4manijold whose symplecticform represents

atr integral cohomology class. Therz,for a sufficiently large k > 0 one can blow-up the manifold

M at d = k2 SW w2 points (see [7,127,213]), so that the resultant manifold (I??, 6) admits a

smooth mup p : M + S’ with the following properties:

a) there exists a ,finite set of points C = (~1. . , z/J, z; E S’, such that pi,, I cs:~,,c, :

p ’ (S’ \ C ) -+ S’ \ C is a jibration with symplectic fibers E 1. . , El ;

b) each exceptional ,fiber E, = p-‘(z;). i = 1. . . . , 1, is un immetsed symplectic surfac~e

M’I th a single transversal double point di E E; :

C) in a neighborhood U, of each of the points d;, i = 1, . . , 1, there exists an integrable

complex structure, compatible with w, and such that the map pin, : Ui + S’ is holomorphic

and d, is a unique, non-degenerate critical point of the holomorphic function p 1 I’, .

It is likely that Theorem 4.3 will play an important role for understanding the topology of

symplectic (4-dimensional?) manifolds. In fact. Theorems 4.2 and 4.3 suggest that symplectic

structures may be more important and interesting than the smooth ones. For instance, the

differential topology in dimension 4 is much richer than the higher-dimensional one. On the

other hand, Theorem 4.2 shows that the symplectic topology of higher-dimensional manifolds

is at least as rich as the symplectic topology of 4-dimensional manifolds.

In dimension 4 the difference between smooth and symplectic structures does not look so

dramatic. For instance, it is possible that the real difference in dimension 4 is not between diffeomorphism and homeomorphism but rather between symplectic and non-symplectic. One

may expect that if two symplectic manifolds are homeomorphic but not diffeomorphic. then actually there is nosymplectic homeomorphism (see ) lo]) between them. This mysterious notion

is yet to be properly defined and understood.

5. Symplectic 4-manifolds

In dimension 4 an alternative technique for finding codimension 2 symplectic submanifolds

is provided by C. Taubes’ discovery of relations between J-holomorphic curves and solutions

to the Seiberg-Witten equations.

72 I: Eliashberg

Notice that in all dimensions a symplectic submanifold is a complex submanifold for a

suitable choice of a compatible complex structure, tamed by the symplectic form. However,

in higher dimension these complex submanifolds are highly unstable: the subspace of almost

complex structures which have any, even local complex submanifolds of complex dimension

> 1 has an infinite codimension in the space of all almost complex structures.

On the other hand all almost complex manifolds have, at least locally, plenty of l-dimensional

complex submanifolds, or (pseudo-)holomorphic curves. It was a major discovery of M. Gromov,

that in the presence of a taming symplectic structure one has a powerful global theory of

holomorphic curves. Because of the positivity of intersections, the holomorphic curves technique

is especially effective in the 4-dimensional symplectic topology.

We begin with the following theorem by Gromov (see [48]) which has already become

classical.

Theorem 5.1. Let (M, w) be a connected symplectic 4-manifold which coincides with the

standard symplectic iR4 outside a compact set. Then if M contains no symplectic 2-spheres (for

instance if nz(M) = 0) then (M, w) is symplectomorphic to the standard symplectic R4.

D. McDuff extended (see [ 1251) this theorem to the case when symplectic 2-spheres may be

present. In this case (M, o) is symplectomorphic to R4 with a few points blown up. Gromov’s theorem, together with McDuff’s extension can be reformulated as follows:

A closed symplectic 4-manifold, which contains a symplectic 2-sphere with self-intersection

index +l, is symplectomorphic to @ P2 with the standard (Fubini-Study) symplectic form,

possibly blown up at a few points.

Removing the assumption about the existence of a symplectic 2-sphere with the self-

intersection index +1 had seemed to be out of reach of the methods of Symplectic Topology

until Taubes found a link between the theory of holomorphic curves and the Seiberg-Witten

theory.

We cannot discuss in the framework of this survey neither Seiberg-Witten theory, nor the

subtle precise definition of Gromov invariants counting the number of holomorphic curves, which is needed for the complete formulation of Taubes’ result. Thus we restrict ourself only to

some corollaries of Taubes’ theory related to the problem of existence of holomorphic curves,

and refer the reader to Taubes’ papers [134,135,136,133] and McDuff’s survey [15] and the

bibliography therein for the detailed account of the story.

In what follows we use the same notations for a homology class and its Poincare dual

cohomology class. The distinction should be clear from the context. Given a symplectic manifold

(M, w) we denote by K its canonical class -q(T(M), J), where J is any almost complex

structure compatible with w, and cl (T(M), J) is the first Chern class of the complex bundle

(T(M), J). Here are two theorems of Taubes (see [ 134,135]).

Theorem 5.2. Let (M, w) be a closed symplectic 4-dimensional manifold with bz > 1 (bz is the positive index of the intersection form of the 4-manifold M). Suppose that K # 0. Then for

a generic almost complex structure J compatible with o

Symplectic topology in the nineties 73

a) the homology class dual to the canonical class K can be represented by an embedded

halomorphic curve;

b) if a homology class A E Hz(M) with A2 = - 1 and K . A # 0 can be represented b>

an embedded 2-sphere then either A, or -A can be represented by an embedded holomorphic

sphere.

Theorem 5.3. The complex projective space @ P2 (considered as a smooth manifold) endowed

with a generic almost complex structure compatible with an arbitrary symplectic structure w

ha., an embedded 2-sphere with the self-intersection index + 1.

Theorem 5.3, together with Gromov’s Theorem 5.1, implies uniqueness (up to diffeomor-

phism) of a symplectic structure on @. P2 compatible with the standard smooth structure.

For symplectic manifolds with b2f = 1 and 61 = 0 with the help of the “wall crossing formula”

of Kronheimer-Mrowka (see [ 1201, and also [16,126.1 for applications) Taubes’ theory implies:

Theorem 5.4. Let (M, w) be a symplectic 4manifold with bt = 1 and bl = 0. Then for a

generic compatible almost complex structure J, and any homology class A E Hz(M), A #

0, K, for which d(A) = $ (A. A - K . A) 3 0, one of the classes A, or K - A can be represented

by a holomorphic curve C passing through any given d(A) points of M.

Theorem 5.4 can be generalized to a larger class of symplectic manifolds, of so-called non-

simple Seiberg-Witten type. The generalization requires a more advanced wall-crossing formula

(see [129,123,130]).

For a generic J, a symplectic manifold M has no connected J-holomorphic curves of negative

self-intersections, except for smooth holomorphic spheres with self-intersection - 1. These

spheres can always be blown down (see [125]), so one can assume that the manifold M is

minimal in the sense that it does not have any holomorphic curves of negative self-intersection.

In this case, we have the following result, see [ 151.

Theorem 5.5. For a generic choice of d(A) points, the holomorphic curve C provided b)

Theorem 5.4 is a smooth curve which consists of several, possibly multiply covered, disjoint

components Cj, i = 1, . . . , p, with Cf 3 0. Multiply covered components can only be tori.

If C2 > 0 then p = 1, i.e., the curve C is irreducible. If C2 = 0 then C may have several

components A,, i = 1, . . , k, with A: = 0. All these components Ai are either spheres

representing the same homology class (and in this case M is a ruled sur&ace), or aE1 the A, ‘s

are tori, representing proportional homology classes.

Due to efforts of several mathematicians the theory of holomorphic curves in symplec-

tic 4-manifolds found a lot of new applications, in particular for the topology of symplectic

4-manifolds and contact 3-manifolds (see [68,118,117] et al.). Let me recall that M. Gro- mov proved in 1481 that the group Diff, of symplectomorphisms of S* x S2 with the split

form w = CJ @ 0, where (T is an area form on S2, deformation retracts to the subgroup of

orientation-preserving orthogonal transformations. It was also indicated in [48] that this re- sult is no longer true when the factors have different symplectic area. M. Abreu (see [115])

14 Y Eliashherg

and M. Abreu-D. McDuff (see [ 1161) completely described the homological type of the group

Diff,(S2 x S2) for the general symplectic form w.

D. McDuff and F. Lalonde used Taubes’ and Gromov’s theorems to obtain a complete clas-

sification of symplectic structures on ruled and rational symplectic manifolds (see [125] and

[ 1221). D. McDuff’s classification of symplectic structures on rational surfaces implies the fol-

lowing striking result (see [ 126,127], and also Biran’s and Lalonde’s papers [201] and [ 12 11).

For positive real numbers ~1, . . . , rk we denote by Emb(ri , . . . , Q) the space of symplectic

embeddings of the disjoint union of balls B(ri), . . . , B(Q) of radii rr , . . . , rk into the unit ball

B(1) c Et” with the standard symplectic structure.

Corollary 5.6. For any positive real numbers rl , . . . , rk the space Emb(rt , . . . , rk) is con-

nected.

Note that for certain choices of r1, . . . , rk the space Emb(ri , . . . , rk) can be empty (see [48]

and [ 1281).

This result sounds to me as counter-intuitive. Indeed, the non-squeezing theorem of Gro-

mov (see Corollary 1.4 above) and different packing inequalities (see [48,128] et al.) lead to

believe that symplectic embeddings of balls behave more like isometric embeddings, than vol-

ume preserving ones. On the other hand, an analog of 5.6 is obviously wrong for isometric

embeddings.

Using Taubes’ theory together with Donaldson’s theory described in Section 4, P. Biran

obtained a nearly complete solution to the symplectic packing problem (see [199] and [200]).

6. Generating functions and their applications

We will discuss in this section a finite-dimensional approach, called the method of gener-

ating functions, which in certain cases allows us to get results which seemed to be currently

unaccessible by holomorphic methods.

6.1. The direct image construction

The direct image construction, which we describe below, is a partial case of a Lagrangian

correspondence, see [3 1,187].

As it was mentioned above, a function f : A4 + R generates an exact graphical Lagrangian submanifold Lf c T*(M) and a graphical Legendrian submanifold Lcf c J’(M).

Given a smooth map h : M + N, one can define, under certain transversality assumptions,

the direct-image construction which allows us to transport Lagrangian submanifolds of T*(M)

into immersed Lagrangian submanifolds of T*(N), and Legendrian submanifolds of J’(M) into Legendrian submanifolds of J1 (IV). In the Lagrangian case, for instance, it can be done

as follows. Consider the manifold W = T*(M) x T*(N) with the symplectic structure fi =

-dp A dq + dfi A d@. Set

H = Wwh)*P, q, 13, h(q)) ; q E M, jj E T;&V) ) .

Symplectic topology in the ninetirs 75

Then H is a Lagrangian submanifold of (W, R). Suppose that for a Lagrangian submanifold

L c T*(M) the product L x N c T*(M) x T*(N) = W is transverse to H. It is then

straightforward to see that the restriction of the projection n : W -+ T*(N) to L f’ H is a

Lagrangian immersion L ~7 H -+ T*(N). The image & = rr(L. n H) c T*(N) is the required direct image of L.

If L = L,, is a graphical Lagrangian submanifold in T*(M), i.e.,

then we say that &, is a subgraphical Lagrangian submanifold generated by the function ,f with respect to the map k : M + N. When k is a diffeomorphism then & = k,(L), where

k, : T*(M) -+ T*(N) is the symplectic lift of the diffeomorphism k, as in Section 1.1 above.

The same holds when k is an equidimensional immersion. When k : A4 + N is an embedding

(or immersion) into a manifold of a larger dimension, then &h can be described as follows.

Set M’ = k(M) c N and consider the restriction map pr : T*(N)] MS -+ T*( M’). Then

L,,, = pr-‘(k,(L)), where k, : T*(M) + T*(M) is the symplectomorphism induced by the

embedding k.

The most interesting examples of the direct image construction are when k : M 4 N is a

submersion and, in particular, a fibration. Suppose, for instance, M = N x F and k : A4 ---f N

is the projection to the second factor. Let (q, r), q E N. q E F, be coordinates in M, and (p, [)

dual coordinates in the cotangent bundle, so that the canonical symplectic form in T*(M) is

given by the form dp A dq + d< A dq. Let L = L, be a graphical Lagrangian submanifold. In

the coordinates (q, q, 4, <) the manifold L is defined by the equations

I ,=h,= af (9, V)

L= a4 ’ ,& =f, Ix ";;?

Then the subgraphical submanifold &, in T*(N) is defined by the equations

Let us consider another extreme case of the direct image construction. Suppose that we are still in the situation when A4 = N x F and k : A4 -+ N is the projection to the second

factor. Notice that T*(M) symplectically splits into the product T*(N) x T*(F). Suppose that a Lagrangian submanifold L c T*(M) is the product L = L 1 x Lz, where L 1, L2 are Lagrangian

submanifolds of T*(N) and T*(F), respectively, and L2 intersects the zero-section in T*(F)

transversally at k points. Then &h consists of k copies of L 1. In particular, when F = R” and

L2 is a Lagrangian plane in T*(l@) = IR2k, transversal to the zero-section, then &, = L 1. In the language of generating functions this can be expressed as follows. Suppose that L = L f is

a graphical Lagrangian submanifold in T*(M) and Q : I& + R a non-degenerate quadratic

form. Given a map k : M + N, the subgraphical submanifold & c T*(N) generated by the

function ,f with respect to the map k coincides with the subgraphical submanifold generated

76 I: Eliashberg

by the function f @ Q : M x I@ += IR with respect to the composition M x I@ ‘T’ M 3 N.

The definition of the direct image construction in the Legendrian case is similar.

6.2. Quadratic stabilization

To make functions on non-compact manifolds amenable to Morse theory we will assume that

all considered functions arejibrations at injnity (see [7,187]).

A function f : M + IR is called a fibration at infinity, if there exist a finite segment

[-a, a] c IR and a compact subset K c f -’ [-a, a] c M such that the restriction of f to the

following three subsets$bers them over their respective images

(i) f-I(--00, -a] + (--00, -a],

(ii) f-‘[a, w) + [a, w), (iii) (f-‘[-a, a]) - K -+ [-a, a].

Let us fix a smooth map h : M + N and a fibration at infinity f : M -+ Et. Given a

Lagrangian submanifold L c T*(M) (or a Legendrian submanifold L c .I' (M)) we denote

by 3(L, h, f) (resp. 3(L, h, f)) the space of all functions f : M + IR which coincide with f

at infinity and transversally generate L (resp. L;) with respect to h. Let us stabilize the spaces

3(L, h, f) and S(ic, h, f) as o f 11 ows. Consider a sequence of spaces 3(L, hk, f CB Qk) and

3;((L, hk, f CD Qk), k = 1, . . . , where

Qk(v) = ~1~2 + . . . + v2k-1~2k, rl = (rll, . . . , r2d E R2?

is a non-degenerate quadratic form on R2k, and hk is the composition M x R2k ‘5’ M 3 N. Now

take the direct limits Ft (L , h , f) and 3”‘(L, h , f) under the inclusions

and similarly in the Legendrian case, defined by the above described stabilization construc-

tion.

6.3. The covering homotopy property

We denote by Lag = kag(N, Lo) the space of exact embedded Lagrangian submanifolds of

T*(N), which coincide with a fixed Lagrangian submanifold Lo at infinity. Let us set

and denote by Gen the projection Flag -+ Lag which associates with a function from F&g the

Lagrangian submanifold from Lag which it generates.

Similarly we define spaces Fzekg = FLk,(&, h, f) andkeg = .&eg(N, &a) andtheprojection

Gen : FLeg + Leg in the Legendrian case. The following Theorem 6.1 (see [185,194,192,186,196,195,187]) is the main result of the

theory of generating functions.

Symplectic topology in the nineties 71

Theorem 6.1. The maps Gen : F;& -+ /Zag and Gen : 3& + Leg are Serrejibrations over

connected components which intersect the image of the map Gen.

In particular, if a Legendrian submanifold La can be generated by a function .f : M -+ IFt with respect to a map h : M + N, and a Legendrian submanifold ,!Z is isotopic to & via a compactly

supported Legendrian isotopy then L can be generated by a function g : M x JR2k -+ IR with

respect to the map hk : M x RI; + N, such that g coincides with f @ Qk at infinity.

Here are few corollaries of Theorem 6.1. Suppose that N is a closed manifold. Let us denote

by stabMor(N) and stabLuS(N) the stable Morse and Lusternik-Shnirelman numbers of the

manifold N. These are the minimal number of critical points of a function f(x. y), x E N. y E

R2k. k = 0 . . , which is equal to Qk(y) outside a compact set. In the Morse case, the critical

points are required to be non-degenerate. The lower bounds on the numbers stabMor(N) and

stabLuS(N) in terms of topology of N is the subject of stable Morse-Lusternik-Shnirelman

theory (see [23 l] and [ 1871). The most commonly used bounds are the Morse inequality

stabMor(N) 3 rank H,(N),

and the Lusternik-Shnirelman inequality

stabLuS (N) 3 cuplength( N) .

However, when N is not simply-connected these inequalities can be essentially improved. See

[23 11 and [ 1871 for discussions of the coresponding results.

Let us denote by Lo the zero-section in T*(N). Suppose that L is an exact Lagrangian sub-

manifold of T*(N) which is Lagrangian isotopic to Lo. We denote by fl(L, Lo) the cardinality

of the intersection L n Lo, and by h(L, Lo) the same number in the case when the intersection

is transversal.

Then Theorem 6.1 implies

Corollary 6.2.

m( L, Lo) 3 stabMor(N) ;

n(L, Lo) 3 stabLuS(N)

The corresponding bounds in terms of Betti numbers and the cup-length constitute one of

Arnold’s symplectic conjectures and were first proven by M. Chaperon (for the case of N = T*“,

see I 185]), F. Laudenbach and J.-C. Sikorav (see [ 1921) and H. Hofer (see 1921). Corollary 6.2 uses only the existence of generating functions, but not the full strength of

Theorem 6.1. The next Corollary 6.3 uses the multi-parametric part of this theorem.

Suppose that V = N x IR for a closed manifold N and denote by n the projection V =

N :x: IR ---f R. Let /Zag, be the space of exact Lagrangian submanifolds of T*(V) which

are isotopic via a compactly supported Lagrangian isotopy to the graphical submanifold L,.

Similarly, the notation Leg, stands for the space of Legendrian submanifolds of J’ (V) which

are Legendrian isotopic to L, via a compactly supported Legendrian isotopy. Any Lagrangian submanifold from Lag, uniquely lifts to a Legendrian submanifold from Leg,. This defines an inclusion i : Lag, Ls Leg,.

78 It Eliashberg

Let 3’(N) be the identity component of the pseudoisotopy group of N, i.e., the group of

diffeomorphisms V -+ V, which preserve the function n outside of a compact set, and which

are equal to the identity on N x (--00, -11. The group P(N) acts on the space Cage, by lifting

diffeomorphisms of V to symplectomorphisms of T*(V) (see Section 1.1 above). We denote

by j the inclusion of ‘P(N) into Lag, as the orbit of L, .

Corollary 6.3. The inclusions j : Y(N) -+ Lag, and i o j : Y(N) --+ Leg0 are injective on

the homotopy groups of 3’(N) of dimension < $n - 2.

This corollary (see [187]), together with known information about the homotopy type of

pseudoisotopy spaces provide non-trivial elements in homotopy groups of spaces of Lagrangian

and Legendrian embeddings. See [187] for a detailed discussion of the subject, as well as for

other applications of the method of generating functions.

7. Old and new open problems

In this section I will review the status of some basic open problems which were discussed in

my survey [5]. As the reader can observe, despite all the progress most of the basic problems

remain wide open.

7.1. Existence and uniqueness of contact and symplectic structures

In dimension > 4 we still do not have any counter-examples to the following “soft conjec-

tures”:

1. Does any odd-dimensional manifold with a stable almost complex structure have a contact

structure?

1’. Does any contact structure, which is given near the boundary of the ball B2nf’ and which

extends to the ball as a stable almost complex structure, extend to the ball as a contact structure?

(A similar symplectic question has the negative answer in all dimensions > 2). Does any closed

odd-dimensional, stably almost complex manifold admit a contact structure?

2. Does any closed 2n-dimensional, 2n > 4, almost complex manifold M with a cohomology

class u E H2(M; IR) with u” # 0 has a symplectic structure (in the cohomology class u)?

3. Is any symplectic, or contact structure on IF, which is standard at infinity, isotopic to the

standard contact structure via a compactly supported isotopy?

The answer to Question 2 is negative in dimension 4. Indeed, a theorem of Taubes (see

[ 13 11) implies that a symplectic 4-manifold cannot split into a connected sum of two manifolds with bz > 0. In particular, @P2#CP2#@P2 has no symplectic structure despite that it has an

almost complex structure and a 2-dimensional cohomology class u with u2 # 0. The answer to Question 3 is “yes” in dimension 4 ([48]) and “no” in dimension 3 ([44]), but “yes” for tight

contact structures in dimension 3 ([%I).

Svmplectic topology in the nineties 79

7.2. Lagrangian and Legendrian embeddings

4. Is any exact embedded Lagrangian submanifold L c T*(M) Hamiltonian isotopic to the O-section? In case M and L are non-compact we assume that M and L coincide at infinity and

want the isotopy to be compactly supported.

5. Let f : S” + S” be a diffeomorphism, non-isotopic to the identity. Let i and ,j be the

inclusions of S” (as the O-sections) into T*(Y) and J’(S”), respectively. Is the Lagrangian

(resp. Legendrian) embedding i o f : S” -+ T*(M) (resp. j o f : S” + J’(M)) Lagrangian

(resp. Legendrian) isotopic to the corresponding inclusion? Notice that i o f and ,i o .f’ are

isotopic to i and j as smooth embeddings.

5’. More globally, let S” c S *w’ be the Legendrian sphere obtained by intersecting S’“+’

with the Lagrangian plane IV+’ c C+’ , and k : S” LIP S2n+’ be the inclusion. Let f : S” + S”

be as in Problem 5. Are j and the composition k o f : S” --+ S2nS-’ Legendrian isotopic?

Problems 4, 5 and 5’ are related to the following question which I first heard from Jeff Mess about 9 years ago.

6. Let C” be an exotic sphere. Are T*(Y) and T* (C”) symplectomorphic?

Some progress towards this problem was achieved in [206].

7.3. Topology of the groups of symplectic and contact diffeomorphisms

Let DD, denote the group of compactly supported symplectic or contact (depending on the pairity of the dimension) diffeomorphisms of the standard symplectic or contact space IRn . For

n < 4 the group ‘_D,, is contractible. For n = 2 this is, essentially, a theorem of S. Smale (see

[233]), for n = 3 this is proven in [55] and for n = 4 this is a result of M. Gromov, see [48].

7. Suppose that n > 4. Is the group 2), contractible?

Nothing is known about the topology of the group ‘_Dn for n > 4. However it is likely that

the methods of [ 1871 may provide non-trivial elements in higher homotopy groups of ‘D,, in the

contact case.

Let Diff, denote the group of compactly supported diffeomorphisms of II??.

8. What are the homotopical properties of the inclusion Dfl c-, Diff,?

For n < 3 this inclusion is a homotopy equivalence. But already in the 4-dimensional case

nothing is known. In view of Gromov’s result for 94 Problem 84 is equivalent to the problem about the topology of the group Diff,. Notice that according to theorems of J. Moser and J. Gray

(see 1.1 above) the factor-space Diff,/YD), is homeomorphic to the space of symplectic forms on I&“, which are standard at infinity.

8. Other developments

In this section we just mention few other important developments during the last decade.

After a sequence of successive improvements Arnold’s conjecture about the number of fixed points of a Hamiltonian symplectomorphism is now proven for a general symplectic manifold,

80 Y: Eliashberg

at least as far it concerns with a lower bound by rational Betti numbers. The final step was made

independently by several groups of authors: K. Fukaya and K. Ono, G. Liu and G. Tian, H. Hofer

and D. Salamon. (see [90,96,94]). K. Fukaya and K. Ono were the first who announced the

solution.

F. Laudenbach partially realized his “engulfing program” (see [215]).

A lot of progress has been achieved towards the Weinstein conjecture about periodic orbits

of a Hamiltonian system (see [176,166,171,172,173]). H. Hofer adapted the technique of

holomorphic curves in compact symplectic manifolds, or manifolds with finite geometry at

infinity (see [48]), for use in symplectizations of contact manifolds (see [75,74]). This technique

has proven to be extremely powerful and useful for results about the Weinstein conjecture and

many other applications. For instance, in the 3-dimensional case it allowed one to go significantly

further in understanding the topology of contact manifolds (see [72,76,77,78,79,9, SO]).

Holomorphic curves in symplectizations were used by Hofer and the author [73] for con-

structing new invariants of contact manifolds and their Legendrian submanifolds, called contact

homology theory. For the case of Legendrian knots in R3 a similar theory was independently

developed by Yu. Chekanov [50] via a purely combinatorial approach.

A new phenomenon of unknottedness of Lagrangian submanifolds of symplectic 4-manifolds

(camp. Problem 4 in the previous section) was discovered in the works of L. Polterovich and

the author, and K. Luttinger (see [ 180,179,18 1,182,184]). Recently H. Hofer and K. Luttinger (unpublished) obtained new strong results in this direction using Hofer’s method of holomorphic

curves in symplectizations. Yu. Chekanov [ 1771 found new invariants of Lagrangian tori in Jk2n

with respect to Hamiltonian isotopy.

R. Gompf (see [209]), rediscovered Gromov’s fibered connected sum construction and trans-

formed it into a powerful machine for constructing symplectic 4-manifolds with prescribed properties. This construction, together with its generalization by J. McCarthy and J. Wolfson

(see [217,216]) and M. Symington (see [234]) were used to disprove many too optimistic

conjectures in 4-dimensional symplectic topology (see [210,211,235]).

V.I. Arnold suggested, and partially proved a symplectic-geometric conjecture generalizing

the classical 4-vertices theorem in Differential Geometry. This generated a lot of works studying

Vassiliev type invariants of Legendrian and Lagrangian wave fronts and caustics (see [ 1971 and the bibliography therein). Despite a lot of interesting new results in this direction Arnold’s

conjecture generalizing 4-vertices theorem still remains wide open. Besides the results discussed in Section 5 there were several other exciting developments in

the theory of holomorphic curves in symplectic manifolds. Let us mention here the foundations

of the theory of Gromov (and Gromov-Witten) invariants, quantum cohomology theory and its

relations with enumerative Algebraic Geometry and mirror symmetry. These led recently to a

partial solution of the Mirror conjecture by A. Givental (see [143,140,142]), see also Lian- Liu-Yau’s preprint [ 1481. P. Seidel found a remarkable application of quantum cohomology for

his study of the effect of the generalized Dehn twist (see [loo]). Despite the large number of references, the bibliography below is far from being complete,

especially in those areas of symplectic topology which are not discussed in this survey. However, the size of the bibliography below should give the reader an idea of the intensity of research in

this field.

Symplectic topology in the nineties Xl

References

Books and surveys

[I] B. Aedischer, M. Borer, M. Kalin, Ch. Leuenberger and H.M. Reimann, Symplectic Geometry. An Introduction

Bused on the Seminur in Bern, 1992 (BirkhBuser, 1994). [2] V.I. Arnold, First steps in symplectic topology, Russian Math. Surveys 41 (1986) l-21. 131 V.I. Arnold and A.B. Givental, Symplectic geometry, in: Dynamical Systems, Vol. 4 (Springer, 1988). [4] M. Audin and J. Lafontaine, eds., Holomorphic Curves in Symplectic Geometry (Birkhtiuser, 1994).

[.‘;I Y. Eliashberg, Three lectures on symplectic topology in Cala Gonone, in: Conference on Differential Geometry and Topology (Sardinia, 1988), Rend. Sem. Fuc. Sci. Univ. Cugliari 58 (1988), suppl., 2749.

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

[7] M. Gromov, Partial DifSerentiul Relations (Springer, 1986). [X] H. Hofer, Symplectic invariants, Proceedings of the International Congress of Muthemuticiuns. Kwto. 1990.

Vol. II (Math. Sot. Japan, Tokyo, 1991) 521-528. 19) H. Hofer and M. Kriener, Holomorphic curves in contact dynamics, Preprint, 1997.

[I I)] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics (Birkhauser, 1995). [I I] C. Hummel, Gromov’s Compuctness Theorem for Pseudo-holomorphic Curves (Birkhluser, Basel, 1997). [I?] F. Lalonde, J-holomorphic curves and symplectic invariants, in: Gauge Theory and Symplecfic Geometn:

Montreal, PQ, 1995, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 488 (Kluwer Acad. Publ.. Dordrecht,

1997) 147-174. [ I.31 D. McDuff and D. Salamon, Introduction to Symplectic Topology (Oxford University Press, New York, 1995). [ 141 D. McDuff, Symplectic 4-manifolds, Proceedings of the International Congress of Mathematicians, Kwto.

lY9U, Vol. II (Math. Sot. Japan, Tokyo, 1991) 541-548. [l-i] D. McDuff, Lectures on Gromov invariants for symplectic 4-manifolds, in: Gauge Theory and Symplectic,

Geometry Montreal, PQ, 1995, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.. 488 (Kluwer Acad. Publ., Dordrecht, 1997) 175-2 10.

[ Ici] D. McDuff and D. Salamon, A survey of symplectic 4-manifolds with h2f = I, Turkish J. Muth. 20 ( 1996)

47-60.

[ 1’71 D. McDuff and D. Salamon, J-holomorphic Curves and Quantum Cohomology, University Lecture Series 6

(American Mathematical Society, Providence, RI, 1994). [IX] D. Salamon, Lectures on Floer homology, Preprint, 1997. [ 191 C.B. Thomas, ed., Contuct and Symplectic Geometr?/ (Cambridge Univ. Press, Cambridge, 1996). [20] C.B. Thomas, 3-dimensional contact geometry, in: Contucr und Symplectic Geometv, Cambridge, 1994, Publ.

Newton Inst. 8 (Cambridge Univ. Press, Cambridge, 1996) 48-65.

Before the birth of Symplectic Topology

[2 I] V.I. Arnold, Sur une propriCtC topologique des applications globalement canoniques de la mCchanique clas- sique, C. R. Acud. Paris 261(1965) 37 19-3722.

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

l-14. [23] V.I. Arnold, Normal forms of functions near degenerate critical points, Weyl groups Ak, @,., Ek and La-

grangian singularities, Funk Anal. i Pril. 6 (1972) 3-25. [23] V.I. Arnold, Lagrange and Legendre cobordism, Funct. Anal. and Applic. 14 (1980) 167-177, 252-260.

[25] A. Banyaga, Sur la structure de la groupe des diffeomorphismes qui prCservent une forme symplectique. Comment. Muth. Helv. 53 (1978) 174-227.

[26] E. Bedford and B. Gaveaux, Envelopes of holomorphy of certain 2-spheres in @*, Amec J. of Muth. 105 (1983) 975-1009.

[27] G.D. Birkhoff, Proof of PoincarC’s geometric theorem, Trans. AMS 14 (1913) 14-22.

[28] E. Calabi, On the group of automorphisms of a symplectic manifold, in: Problems in Analysis (Princeton Univ. Press, 1970) l-26.

[29] J.W. Gray, Some global properties of contact structures, Annals of Math. 69 (1959) 421450.

82 l! Eliashberg

[30] M. Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Proc. KM-70, Nice, Vol. 2 (Gauthier-Villas, Paris, 1971) 221-225.

[31] L. HGrmander, Fourier integral operators I, Acta Math. 127 (1971) 79-183. [32] C.G.J. Jacobi, Vorlesungen iiber Dynamik (1842-1843) (G.Reimer, Berlin, 1884).

[33] R. Lutz, Structures de contact sur les fibrCs principaux en cercles de dimension 3,Ann. Inst. Fourier 27 (1977)

l-15.

[34] R. Lutz, Sur la gCometrie des structures de contact invariantes, Ann. Inst. Fourier 29 (1979) 283-306. [35] J. Martinet, Formes de contact sur les variCtCs de dimension 3, Lecture Notes in Math. 209 (197 1) 142-163.

[36] J. Moser, On the volume elements of a manifold, Truns. Amer: Math. Sot. 120 (1965) 286-294. [37] A. Poincark, Sur une thCor&me de gComCtrie, Rendiconti Circolo Mat. Palermo 33 (1912) 375-507.

[38] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978) 157-184.

[39] P. Rabinowitz, Periodic solutions of Hamiltonian systems on a prescribed energy surface, J. Difl Eq. 33 (1979)

336-352. [40] W.P. Thurston, Some simple examples of symplectic manifolds, Proc. AMS 55 (1976) 467468. [41] A. Weinstein, On the hypotheses of Rabinowitz’s periodic orbits theorems, J. DQj! Eq. 33 (1979) 353-358. [42] A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Ann. Math. 98 (1973) 377-410

[43] A. Weinstein, Lectures on Symplectic Geometry, CBMS Reg. Conf. Series in Math. 29 (AMS, Providence,

R.I., 1979).

Emergence of Symplectic Topology

[44] D. Bennequin, Entrelacements et Cquations de Pfaff, Aste’risque 106-107 (1983).

[45] C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold, Invent.

Math. 73 (1983) 3349.

[46] Y. Eliashberg, Estimates of the number of fixed points of area-preserving transformations of surfaces, Syk- tyvkar University, 1978 (Copyright VINITI, 1979) l-105.

[47] Y. Eliashberg, The theorem on the structure of wave fronts and its applications in Symplectic geometry, Funct. Analysis and it Applic. 21 (1987) 65-72.

[48] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307-347.

Contact 3-manifolds

[49] S. Altschuler, A geometric heat flow for one-forms on three-dimensional manifolds, Illinois J. ofMath. 39

(1995) 98-l 18. [50] Yu. Chekanov, Differential algebra of a Legendrian link, Preprint, 1997. [5 l] V. Colin, Chirurgies d’indice un et isotopies de sphCres, Preprint, 1996. [52] V. Colin, Recollement des variCtCs de cotact tendues, Prepint, 1997. [53] V. Colin, Stabilitk topoloqique des structures de contact en dimension 3, Preprint ENS Lyon N. 225, 1998.

[54] Y. Eliashberg, Classification of overtwisted contact structures, Invent. Math. 98 (1989) 623-637. [55] Y. Eliashberg, Contact 3-manifolds twenty years after J. Martinet’s work, Annales de l’lnst. Fourier 42 (1992)

165-192.

[56] Y. Eliashberg, Unique holomorphically fillable contact structure on the 3-torus, Internat. Math. Res. Notices 1996 (2) 77-82.

[57] Y. Eliashberg, Legendrian and transversal knots in tight contact manifolds, in: Topological Methods in Modern

Mathematics (Publish or Perish, 1993). [58] Y. Eliashberg, Classification of contact structures on R3, Int. Math. Res. Notices 3 (1993) 87-91. [59] Y. Eliashberg, Filling by holomorphic discs and its applications, London Math. Sot. Lect. Notes Ser: 151

(1991) 45-67. [60] J. Etnyre, Tight contact structures on Lens spaces, Preprint, 1997. [61] H. Geiges and J. Gonzalo, Topology and Analysis of contact circles, in: Contact and Symplectic Geometry

(Cambridge Univ. Press, 1996) 66-77. [62] H. Geiges, Normal contact structures on 3-manifolds, T6hoku Math. J. 49 (1997) 415-422.

[63] E. Giroux, ConvexitC en topologie de contact, Comm. Math. Helvet. 66 (1991) 637-677.

Symplectic topology in the nineties 83

[64] E. Giroux, Une structure de contact, m&me tendue est plus ou moins tordue, Ann. Scient. EC. Norm. Sup. 27

( 1994) 697-705.

[65] E. Giroux, Structures de contact sur les fibrCs aux circles, Preprint, 1994. [66] R. Gompf, Handlebody constructions of Stein surfaces, Preprint, 1996; Annuls ofbfuth., to appear. [67] Y. Kanda, The classification of tight contact structures on the 3-torus, Preprint, 1995. [6X] P. Kronheimer and T. Mrowka, Monopoles and contzt structures, Invent. Muth. 130 (1997) 209-255. [69] S. Maker-Limanov. Tight contact structures on 3-dimensional solid tori, Truns. Ame,: Muth. SW. 350 ( 1998)

10!3-1044. [70] S. Makar-Limanov, Morse surgeries of index 0 and I on tight manifolds, Preprint, 1996.

[7 I ] P. Lisca, Symplectic fillings, and positive scalar curvature, Preprint, 1998.

Holomorphic curves in contact geometry

[7.1] Y. Eliashberg and H. Hofer, A Hamiltonian characterization of the three-ball, Differential Integrul Equnrions

7 (1994) 1303-1324.

[7 3] Y. Eliashberg and H. Hofer, Contact homology theory, in preparation. [74] Y. Eliashberg, H. Hofer and D. Salamon, Lagrangian intersections in contact geometry, Geom. Arnct. Anul.

5 (1995) 244-269. [75] H. Hofer, Pseudo-holomorphic curves and Weinstein conjecture in dimension three, Invent. Muth. 114 ( 1993)

515-563. [76] H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations, I.

Asymptotics. Ann. Inst. H. Poincuri, Anaf. Non Liniaire 13 (I 996) 337-379.

[77] H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations, II. Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995) 270-328.

[7X] H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations. III. Fredholm theory, Preprint, 1997.

[79] H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations. IV. Asymptotics with degeneracies, in: Contact and Symplectic Geometry (Cambridge, 19941, Publ. Newton Inst. 8 (Cambridge Univ. Press, Cambridge, 1996) 171-200.

[8O] H. Hofer, K. Wysocki and E. Zehnder, A characterisation of the tight three-sphere, Duke Math. J. 81 (1995) 159-226.

Floer homology theory

[X 1 ] Yu. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Preprint ETH. 1996.

[ 871 A. Floer, Morse theory for Lagrangian intersection theory, J. Difl Geom. 28 (1988) S 13-547.

[X3] ,4. Floer, The unregularised gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988) 775-813.

]%I] A. Fleer, A relative index for the symplectic action, Comm. Pure und Appl. Math. 41 (1988) 393-407.

[X5] A. Floer, Witten’s complex and infinite dimensional Morse theory, J. D$ Geom. 30 (1989) 207-22 I. [Xh] A. Fleer, Symplectic fixed points and holomorphic spheres, Comm. Math. Physics 120 (1989) 576-6 I I [X7] A. Fleer and H. Hofer. Coherent orientations for periodic orbit problems in symplectic geometry, Muth. Z.

212(1993) 13-38. (XX] A. Fleer, H. Hofer and D. Salamon, Transversality in elliptic Morse theory for the symplectic action. /11&e

Muth. J. 80 (1995) 25 I-292.

[X9] B. Fortune and A. Weinstein, A symplectic fixed point theorem for complex projective spaces. Bull. AMS 12 (1985) 128-130.

[SO] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariants, Preprint, 1996. [9 I ] A. Givental, Periodic maps in Symplectic Topology, Fact. Anul. Appl. 23 (1987) 361-390. [W] H. Hofer, Lagrangian embeddings and critical point theory, Ann. Inst. A. PoincarP 2 (1985) (6) 4071162.

1931 H. Hofer and D. Salamon, Fleer homology and Novikov rings, in: The Fleer Memoriul Volume (Birkhguser. 1995) 483-524.

[93] H. Hofer and D. Salamon, to appear.

84 K Eliashberg

[9.5] H. Hofer, Ljustemik-Schnirelman theory for Lagrangian intersections, Ann. Inst. H. Poincare’ 5 (1988) 465-

499.

[96] G. Liu and G. Tian, Floer homology and Arnold conjecture, Preprint, 1997. [97] Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, Part I, Comm. Pure

Appl. Math. 46 (1993) 949-993; Part II, Comm. Pure Appl. Math. 46 (1993) 995-1012; Part III, in: The Fleer

Memorial Volume (Birkhluser, 1995) 555-573. [98] K. Ono, On the Arnold conjecture for weakly monotone symplectic manifolds, Invent. Math. 119 (1995)

5 19-537.

[99] J. Robbin and D. Salamon, The Maslov index for paths, Topology 32 (1993) 827-844. [loo] P. Seidel, Floer homology and the symplectic isotopy problem, PhD thesis, University of Oxford, 1997. [ 1011 A. Weinstein, On extending Conley-Zehnder fixed point theorem to other manifolds, Proc. Symp. Pure Math.,

AMS 45 (1986).

[ 1021 E. Witten, Supersymmetry and Morse theory, J. Diff Geom. 17 (1982) 661492. [103] Y. Ruan and G. Tian, Bott-type symplectic Floer cohomology and its multiplication structures, Math. Res.

Letr. 2 (1995) 203-219.

Hofer geometry

[ 1041 M. Bialy, L. Polterovich, Geodesics of Hofer’s metric on the group of Hamiltonian diffeomorphisms, Duke Math. J. 76 (1994) 273-292.

[ 1051 Y. Eliashberg, L. Polterovich, Bi-invariant metrics on the group of Hamiltonian diffeomorphisms, Internat. J. Math. 4 (1993) 727-738.

[106] H. Hofer, On the topological properties of symplectic maps, Proc. Royal Sot. Edinburgh, Sect. A 115 (1990) 25-38.

[107] H. Hofer, Estimates for the energy of a symplectic map, Comment. Math. Helv. 68 (1993) 48-72.

[108] F. Lalonde and D. McDuff, Hofer’s Lo3 -geometry: energy and stability of Hamiltonian flows. I, II, Invent.

Math. 122 (1995) l-33,35-69.

[ 1091 F. Lalonde and D. McDuff, Local non-squeezing theorems and stability, Geom. Funct. Anal. 5 (1995) 364-

386. [1 lo] F. Lalonde and D. McDuff, The geometry of symplectic energy, Ann. ofMath. 141 (1995) 349-371.

[ 11 l] F. Lalonde and L. Polterovich, Symplectic diffeomorphisms as isometries of Hofer’s norm, Topology 36 (1997) 711-727.

[ 1121 L. Polterovich, Symplectic displacement energy for Lagrangian submanifolds, Ergo&c Theory und Dynamical

Systems 13 (1993) 357-367. [I 131 L. Polterovich, Hofer’s diameter and Lagrangian intersections, Preprint, 1997. [114] I. Ustilovsky, Conjugate points on geodesics in Hofer’s metric, Di# Geom. Appl. 6 (1996) 327-342.

Holomorphic curves in symplectic 4-manifolds

[ 1151 M. Abreu, Topology of the symplectomorphism group of S2 x S2, Invent. Math., to appear. [I 161 M. Abreu and D. McDuff, Topology of symplectomorphism groups of 4-manifolds, Preprint, 1997.

[117] J. Bryan and NC. Leung, The enumerative geometry of K3-surfaces and modular forms, Preprint, 1997. [118] E.-N. Ionel and T. Parker, Gromov invariants and symplectic maps, Preprint, 1997. [119] D. Kotschick, J.W. Morgan and C.H. Taubes, Four-manifolds without symplectic structures but with nontrivial

Seiberg-Witten invariants, Math. Res. L&t. 2 (1995) 119-124. [ 1201 P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Letters 1

(1994) 797-808. [ 1211 F. Lalonde, Isotopy of symplectic balls, Gromov’s radius, and the structure of ruled symplectic 4-manifolds,

Math. Ann. 300 (1994) 273-296. [122] F. Lalonde and D. McDuff, The classification of ruled symplectic 4-manifolds, Math. Res. Lett. 3 (1996)

769-778. [123] T.J. Li and A. Liu, General wall-crossing formula, Preprint, 1995. [124] T.J. Li and A. Liu, Symplectic structure on ruled surfaces and [ 1251 D. McDuff, The structure of rational and ruled symplectic 4-manifolds, J. AMS 3 (1990) 679-712.

Symplectic topology in the nineties 85

[ 1261 D. McDuff, From symplectic deformation to isotopy, Preprint, 1996. [ 1271 D. McDuff, Blow ups and symplectic embeddings in dimension 4, Topology 30 (1991) (3) 409-42 1. [ 1281 D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994) 40%

434; Generalized adjunction formula, Preprint, 1995. [ 1291 H. Ohta and K. Ono, Symplectic 4-manifolds with 1,; = 1, Lect. Notes in Pure and Appl. Math. 184 ( 1997)

237-244.

[130] C. Okonek and A. Teleman, Seiberg-Witten invariants for manifolds with bt = 1, and the universal wall-

crossing formula, Preprint, 1996. [ 13 I] C.H Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Rex Lett. 1 (1994) 809-822. [ 1.221 C.H. Taubes, More constraints on symplectic forms from Seiberg-Witten invariants, Math. Res. Lett. 2 ( 1995)

9-13.

[I 131 C.H. Taubes, Counting pseudo-holomorphic submanifolds in dimension 4, J. Diferential Geom. 44 ( 1996) 818-893.

[133] C.H. Taubes, Seiberg-Witten and Gromov invariants, in: Geometry and Physics, Aarhus, 1995, Lecture Notes in Pure and Appl. Math. 184 (Dekker, New York, 1997) 591-601,

[I iSI C.H. Taubes, SW =+ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer Muth.

Sot. 9 ( 1996) 845-9 18. [ 1361 C.H. Taubes, Gr + SW. From pseudo-holomorphic curves to Seiberg-Witten equation, Preprint, 1996.

Approximately complex submanifolds, Lefshetz fibrations etc.

[ 1371 F. Bogomolov and L. Katzarkov, Symplectic 4-manifolds and projective surfaces, Preprint, 1997.

[ 1381 S. Donaldson, Symplectic submanifolds and almost-complex geometry, J. DifJ: Geom. 44 (1996) 666705. [ 1391 S. Donaldson, Symplectic 4-manifolds and Lefshetz fibrations, to appear.

Quantum cohomology and mirror symmetry

[ IW] A. Givental, A mirror symmetry for toric complete intersections, Preprint, 1997. [ I4 I] A. Givental, Homological geometry and mirror symmetry, in: Proceedings of the International Congress of

Mathematicians at Zurich, Vol. 2 (Birkhauser, 1995) 472-480. [ 1421 A. Givental, Elliptic Gromov-Witten invariants and the generalized mirror conjecture, Preprint, 1998.

[l-13] A. Givental, Equivariant Gromov-Witten invariants, Zntemat. Math. Res. Notices, 1996 (13) 613-663. [ 1341 A. Givental and B. Kim, Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phvs. 168

(1995)609-641. [ 1551 M. Kontsevich, Mirror symmetry in dimension 3, in: Stminaire Bourbaki, Asterisque 237 (1996) 275293.

[I $61 M. Kontsevich, Homological algebra of mirror symmetry, Proc. ICM, 1994, Zurich, Vol. 1 (Birkhluser, 1995) 120-139.

[ 1471 J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer: Math.

Sot., to appear. [ 1181 H. Lian, K. Liu and S.T. Yau, Mirror Principle I, Preprint, 1997. [ 1491 S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology,

in: Contact und Symplectic Geometry, Cambridge, 1994, Publ. Newton Inst. 8 (Cambridge Univ. Press, Cambridge, 1996) 17 l-200.

[ I501 Z. Qin and Y. Ruan. Quantum cohomology of projective bundles over Pn, Trans. Amer Math. Sac.. to appear.

[ I 5 I] Y. Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83 ( 1996) 461-500.

[ 1521 Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 ( 1995) 259-367.

[ 1531 Y. Ruan and G. Tian, Higher genus symplectic invariants and sigma-model coupled with gravity, Invent. Math. 130 (1997) 455-516.

[ 1541 G. Tian, Quantum cohomology and its associativity, in: Current Developments in Mathematics, Cambridge. MA. 1995 (Internat. Press, Cambridge, MA, 1994) 361401.

86 k: Eliashberg

Symplectic topology and Hamiltonian dynamics. Symplectic invariants

[ 1551 M. Bialy and L. Polterovich, Invariant tori and symplectic topology, in: Sinai’s Moscow Seminar on Dynamical

Systems, Amex Math. Sot. Transl. Sex 2 171(1996) 23-33. [156] M. Bialy and L. Polterovich, Hopf-type rigidity for Newton equations, Math. Rex Lett. 2 (1995) 695-700. [157] M. Bialy and L. Polterovich, Optical Hamiltonian functions, in: Geometry in Partial Differential Equations

(World Sci. Publishing, 1994) 32-50. [158] M. Bialy and L. Polterovich, Hamiltonian systems, Lagrangian tori and Birkhoff’s theorem, Math. Ann. 292

(1992) 619-627.

[159] K. Cieliebak, Symplectic boundaries: creating and destroying closed characteristics, Geom. Funct. Anal. 7

(1997). [ 1601 K. Cieliebak, A. Floer and H. Hofer, Symplectic homology, II. A general construction. Math. Z. 218 (1995)

103-122. [161] K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology, II. Stability of the

action spectrum, Marh. Z. 223 (1996) 27-45. [ 1621 I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics I, II, Math. Zeit. 200 (1990) 355-378,

203 (1990) 553-567. [163] Y. Eliashberg and H. Hofer, Towards the definition of symplectic boundary, Geom. Funct. Anal. 2 (1992)

2 1 l-220. [ 1641 Y. Eliashberg and H. Hofer, Unseen symplectic boundaries, in: Manifolds and Geometry, Piss, 1993, Sympos.

Math. 36 (Cambridge Univ. Press, 1996) 178-189. [ 1651 A. Floer and H. Hofer, Symplectic homology, I. Open sets in C”, Math. Z. 215 (1994) 37-88. [ 1661 A. Floer, H. Hofer and C. Viterbo, The Weinstein conjecture in P x Cl, Math. Z. 203 (1990) 469-482. [167] A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology I, Math. Z. 217 (1994) 577-606. [168] V. Ginzburg, A smooth counter-example to the Hamiltonian Seifert conjecture in R6, Int. Math. Rex Not.,

1997 (13) 641-650. [ 1691 H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in: Analysis et cetera (Academic Press,

1990). [ 1701 H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl.

Math. 45 (1992) 583-622.

[171] H. Hofer and C. Viterbo, The Weinstein conjecture in cotangent bundles and related results, Ann. Scuola

Norm. Sup. Piss 15 (1988) 411445 (1989). [172] H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math. 90

(1987) l-9. [173] G. Liu and G. Tian, Weinstein conjecture and GW invariants, Preprint, 1997. [174] L. Polterovich, On strictly ergodic Hamiltonian products, Preprint, 1997. [175] M. Schwarz, A capacity for closed symplectically aspherical manifolds, Preprint, 1997. [176] C. Viterbo, A proof of the Weinstein conjecture in R2” , Ann. Inst. Henri Poincart!, Analyse non Lintfaire 4

(1987) 337-357.

Topology of Lagrangian embeddings

[ 1771 Yu. V. Chekanov, Lagrangian tori in a symplectic vector space and global symplectomorphisms, Math. Z. 223 (1996) 547-559.

[ 1781 Y. Eliashberg and L. Polterovich, New applications of Luttinger’s surgery, Comm. Math. Helvet.

[ 1791 Y. Eliashberg and L. Polterovich, Unknottedness of Lagrangian surfaces in symplectic 4-manifolds, Znt. Math.

Res. Notices., 1993 (11) 295-301. [ 1801 Y. Eliashberg, Symplectic geometry of 2-knots in R4, in: Floer Memorial Volume.

[ 1811 Y. Eliashberg and L. Polterovich, The problem of Lagrangian knots in four-manifolds, in: Geometric Topology, Athens, GA, 1993, AMSnP Stud. Adv. Math. 2.1 (AMS, 1997) 3 13-327.

[182] Y. Eliashberg and L. Polterovich, Local Lagrangian 2-knots are trivial, Ann. ofMath. 144 (1996) 61-76. [183] F. Lalonde and J.-C. Sikorav, Sous-variCtCs lagrangiennes exactes des fib& cotangents, Comment. Math.

Helvet. (1991) 18-33. [ 1841 K. Luttinger, Lagrangian tori in R4, J. D$ Geometry 42 (1995) 220-228.

Symplectic topology in the nineties x7

Generating functions approach

[I 8.51 M. Chaperon, Une idee du type “geodesique brisee” pour les systemes Hamiltoniens, CR. Acad. Sci. Puri.r Sc;r: I Math. 298 (1984) 293-296.

1861 Yu. Chekanov, Critical points of quasi-functions and generating families of Legendrian manifolds, Fuf2c.r. Anal. Pril. 30 ( 1996).

I871 Y. Eliashberg and M. Gromov, Lagrangian intersections theory. Finite-dimensional approach. Preprint 1997. to appear in the Arnold volume.

1 X8] E. Ferrand, Quasi-functions and quasi-hypersurfaces, Preprint, 1996. 1891 E. Ferrand, On a theorem of Chekanov, &much Center Puhlicntions, 1996. 1901 A. Givental, Non-linear generalization of Maslov index, Advances in Soviet Mathematics 1 ( 1990) 7 I -I 03.

[ 19 I] A. Givental, A symplectic fixed point theorem for toric manifolds. The Fleer Memorial Volzlmr (Birkhauser. 199.5) 44548 I.

[ 1021 F. Laudenbach and J.-C. Sikorav, Persistence d’intersections avec la section nulle au tours d’une isotopic

Hamiltonienne darts le fibre cotangent, Invent. Math. 82 (1985) 349-3.57. [ I Y3 I P. Pushkar’, Generalised Chekanov’s theorem. Diameters of of immmersed manifolds and wave fronts. Pre-

print, 1996.

[ lY4] J.-C. Sikorav, Problemes d’intersections et de points fixes en geometric hamiltonienne, Comm. Math. He/l,rt. 62 (1987) 62-73.

[ 1 YS] D.ThCret, A complete proof of Viterbo’s uniqueness theorem for generating functions, Preprint 1997, Top&e> and its Applic., to appear.

[ 1961 C. Viterbo. Symplectic topology as the geometry of generating functions. Math. Annulen 292 ( 1992) 685-7 IO.

Other references

[ 1971 V.I. Arnold, Topological lnvuriants of Plane Curves and Caustics, University Lect. Ser. (AMS, Providence RI, 1994).

[1%X] E. Bedford, W. Klingenberg, On the envelope of holomorphy of a 2-sphere in c*, J. Amer Muth. Sec. 4 (1991).

[ 1 W] P. Biran, Symplectic packing in dimension 4, GAFA 7 (1997) 420437. [2(K)] P. Biran, A stability property of symplectic packing, Preprint, 1997.

[20 I ] P. Biran, Connectedness of spaces of symplectic embeddings, Int. Math. Res. Notices, 1996 ( IO) 48749 1. [2!)2] S. Dostoglou, Stamatis and D. Salamon, Instanton homology and symplectic fixed points, in: Symplectic

Geometry, London Math. Sot. Lecture Note Ser. 192 (Cambridge Univ. Press, Cambridge. 1993) 57-93. [203] Y. Eliashberg. Topological characterization of Stein manifolds of complex dimension > 2. ht. J. of Mnth 1

(1991) 2946. [204] Y. Eliashberg, New invariants of symplectic and contact manifolds, Int. Muth. Rex Notices.

[205] Y. Eliashberg, Symplectic geometry of plurisubharmonic functions, in: Proceedings ofNAT ASfL1YY5 ur Montrkd, to appear.

[206] Y. Eliashberg and M. Gromov, Convex symplectic manifolds, Proc. of Swnp. in Pure Math.. vol.52 ( I991 ). Part 2, 135-162.

[207] D. Gabai, Foliations and the topology of 3-manifolds, .I. D@ Geometq 18 (1983) 445-503. [20X] H. Geiges, Applications of contact surgery, Topology 36 (1997) I 193-I 220. [209] R. Gompf, A new construction of symplectic manifolds, Ann. ofMuth. 142 (1995) 527-595.

12 IO] R. Gompf, Smooth 4-manifolds and symplectic topology, in: Proceedings of the lnternationul Congress of

Mathemuticiun.~, Ziirich, 1994, Vol. 2 (Birkhauser, Basel, 1995) 54%S53. [2 I I] R. Gompf and T. Mrowka, Irreducible 4-manifolds need not be complex, Ann. c!fMuth. 138 (1993) 61-l 1 I. 12 i 21 M. Gromov. J. Lafontaine and P. Pansu, Structures metriques pour les varietes Riemanniennes (Cedic-Femand

Nathan, Paris, 198 I). [2! 31 V. Guillemin and S. Sternberg, Birational equivalence in the symplectic category, Invent. Mnth. 97 ( 1989)

485-522.

[2 141 R. Hind. Filling by holomorphic disks with weakly pseudoconvex boundary conditions, Genm. Funct. AnuI. 7 (1997) 462Wl9.5.

88 Z Eliashberg

[215] F. Laudenbach, Engouffrement symplectique et intersections lagrangiennes, Comment. Math. Helv. 70 (1995) X8-614.

[216] J.D. McCarthy, J.G. Wolfson, Symplectic gluing along hypersurfaces and resolution of isolated orbifold singularities, Invent. Math. 119 (1995) 129-154.

[217] J.D. McCarthy and J.G. Wolfson, Symplectic normal connect sum, Topology, 33 (1994) 729-764. [218] D. McDuff, Examples of symplectic structures, Invent. Math. 89 (1987) 13-36. [219] D. McDuff, Singularities of J-holomorphic curves in almost complex 4-manifolds, .I. Geom. Anal. 2 (1992)

249-266. [220] D. McDuff, Symplectic manifolds with contact type boundaries, Invent. Math. 103 (1991). [221] M.J. Micallef and B. White, The structure of branch points in minimal surfaces and in pseudoholomorphic

curves, Annals ofMath. 141 (1995) 35-85.

[222] A. Nijenhuis and W. Wolf, Some integration problem in almost complex and complex manifolds, Ann. Math.

77 (1963) 424-489.

[223] Y. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internaf.

Math. Res. Notices 1996 (7) 305-346. [224] T.H. Parker and J.G. Wolfson, Pseudoholomorphic maps and bubbling trees, .I. Geom. Anal. 3 (1993) 63-98.

[225] L. Polterovich, Gromov’s K-area and symplectic rigidity, Geom. Funct. Anal. 6 (1996) 726-739.

[226] L. Polterovich, The Maslov class rigidity and nonexistence of Lagrangian embeddings, in: Symplectic Geom-

etry, London Math. Sot. Lecture Note Ser. 192 (Cambridge Univ. Press, Cambridge, 1993) 197-201. [227] Y. Ruan, Symplectic topology and extremal rays, Geom. Funct. Anal. 3 (1993) 395430. [228] Y. Ruan, Symplectic topology and complex surfaces, in: Geometry and Analysis on Complex Manifolds (World

Sci. Publishing, 1994) 171-197.

[229] Y. Ruan, Symplectic topology on algebraic 3-folds, J. Differential Geom. 39 (1994) 215-227. [230] D. Salamon, Lagrangian intersections, 3-manifolds with boundary, and the Atiyah-Floer conjecture, in: Pro-

ceedings of the International Congress of Mathematicians, Ziirich, 1994, Vol. 2 (Birkhluser, 1995) 526-536. [231] V. Sharko, Functions on manifolds. Algebraic and topological aspects, (AMS, Providence RI, 1993). [232] J.-C. Sikorav, Rigidite symplectique dans le cotangent de 7’ a, Duke Math. J. 59 (1989) 227-23 1.

[233] S. Smale, Diffeomorphisms of the 2-sphere, Proc. Ame,: Math. Sot. 10 (1959) 621-626. [234] M. Symington, A new symplectic surgery: the 3-fold sum, Preprint, 1997. [235] Z. Szabo, On irreducible simply-connected 4-manifolds, Turkish J. Math. 21 (1997).

[236] A. Weinstein, Contact surgery and symplectic handlebodies, Hokkuido Math. J. 20 (1991) 241-25 1. [237] R. Ye, Gromov’s compactness theorem for pseudo holomorphic curves, Trans. Amer: Math. Sot. 342 (1994)

67 l-694.