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
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