HOMOLOGY THEORIES IN CONTACT GEOMETRY
UMUT VAROLGUNES - UMUTVG@MIT.EDU
1. introduction
Symplectic Field Theory [1] is a rather general framework for computing invariants of symplectic cobordisms. From SFT, one can extract three homology
theories for contact manifolds; Cylindirical Contact Homology, Contact Homology,
and Legendrian Contact Homology. CCH and CH are invariants that can be used
to distinguish contact manifolds up to contactomorphism, whereas LCH is used to
distinguish Legendrian submanifolds of a fixed contact manifold up to Legendrian
isotopy.
CCH is constructed in a way that is really similar to Floer Homology for symplectic manifolds with a specified Hamiltonian symplectomorphism. SFT essentially arises from first generalizing CCH to CH, by almost purely algebraic means,
and then realizing that the theory would also fit into the more general setting of
symplectic cobordisms. However, we will not get into this any further in this paper.
For LCH, in the case of R3 with the standard contact structure, Chekanov managed to translate partially the geometric framework to a purely combinatorial one
[2]. Later, Etnyre, Sabloff, and Ng completed this translation, and showed that the
two different constructions indeed give the same invariant [4].
There is another homology theory defined for contact manifolds, initiated by
Hutchings, called Embedded Contact Homology, independant of the SFT program
[7]. Even though the construction of ECH is really similar to CH, it turns out to
be a topological invariant. In fact, it is shown by Taubes that ECH is isomorphic
to Seiberg-Witten Floer homology of the underlying manifold [8]. This is in sharp
contrast with the other homologies described, which are really contact invariants.
We will not get into ECH in this paper.
The paper is organized as follows: Section 2 is a background chapter that has
rather sketchy discussions about finite dimensional Morse homology, pseudoholomorphic curves, Conley-Zehnder index, and stable tame isomorphism classes of
differential graded algebras; in Section 3, we construct CCH; in Section 4 we construct CH; in Section 5 we construct LCH both in geometric and the combinatorial
pictures in the case of (R3 , ξst ), and discuss their equivalence.
Here we also note that to avoid the discussion of orientations, which can be quite
complicated, we work over Z2 whenever possible, even though everything in this
paper can be done using Z. The main problem is to orient the moduli spaces that
arise in the constructions, and this is done using the coherent orientations idea of
Floer and Hofer [6]. The author feels sad for not getting into this interesting part
of the theory.
1
2
UMUT VAROLGUNES - UMUTVG@MIT.EDU
2. Background
2.1. Finite Dimensional Morse Homology. In this section, we describe finite
dimensional Morse homology. We also sketch a proof of the fact that it does not
depend on the choices made for the construction directly, without referring to its
equivalence to the singular homology. We believe that these provide important toy
models for the material in the rest of the paper. The reader is referred to [9], for
details.
Let M be a finite dimensional closed smooth manifold and f : M → R be a
Morse function. Given a metric g on M , we can define the gradient flow Φs , s ∈ R
of f . Let us define the following subsets of M , for a critical point p of f ;
(1)
A(p) := {x ∈ M | lim Φs (x) = p}
(2)
D(p) := {x ∈ M | lim Φs (x) = p}.
s→+∞
s→−∞
A(p) and D(p) are embedded open discs with the expected dimensions; dim A(p) =
dim M − ind(p) and dim D(p) = ind(p). For a generic choice of g, the submanifolds
A(p) and D(q) are transversal, for all critical point pairs (p, q) of f . We call such
a pair (f, g) Morse-Smale.
Let M (p, q) be the moduli space of flow lines that tend to p at −∞, and to q
at +∞, modulo the trivial R action by translation. We have the following homeomorphism, which, in particular, lets us compute the dimensions of these moduli
spaces,
M (p, q) = (D(p) ∩ A(q))/R.
(3)
The following intuitive theorem is crucial for the theory. It says that if we also
include the broken flow lines that can be well approximated by the elements of
the moduli space, then we get a compact manifold with corners. Producing such
theorems is really the main technical challenge in the general theory.
Theorem 1. M (p, q) has a natural compactification to a smooth manifold with
corners M (p, q) whose codimension k stratum is
[
(4)
M (p, r1 ) × M (r1 , r2 ) × . . . M (rk , q),
where the union is taken over all critical points r1 , . . . rk which are pairwise distinct
and also distinct from p and q.
Note that as a corollary we deduce that, in case ind(p) − ind(q) = 1, M (p, q) is a
finite set. We are now in a position to define the Morse complex. The ith grade of
the chain complex is defined to be the free module over Z2 with generators in one
to one correspondence with the critical points with index i of f , for every i ∈ Z.
The differential is given by the following formula, where # denotes the number of
points of the zero dimensional manifold modulo 2,
X
(5)
∂(p) =
#M (p, q) · q.
ind(q)=i−1
2
We can check that ∂ = 0 by noticing that the coefficient of q in ∂ 2 is simply given
by the number of the boundary components of the compact 1-manifold M (p, q)
modulo 2. The Morse Homology is then defined to be the homology of this chain
complex. Now, we will sketch why this does not depend on the choice of f and
HOMOLOGY THEORIES IN CONTACT GEOMETRY
3
g. This shows that the Morse homology is a diffeomorphism invariant. It actually
only depends on the homotopy type of the underlying manifold, as it turns out to
be isomorphic to the singular homology.
We first construct a chain map between the Morse complexes corresponding to
different Morse-Smale pairs (f0 , g0 ) and (f1 , g1 ). This is done by taking a generic
path of function and metric pairs {(ft , gt )} between them. Namely, one constructs
the vector field V = t(t − 1)(t + 1)∂t + Vt , where Vt is the gradient flow of the pair
(ft , gt ), on M × I. We then define a Morse complex, using V in the place of the
gradient vector field, in the same way described above. The critical points of V are
in the top and bottom levels only, corresponding to the critical points of f0 and
f1 , with the indices of the bottom critical points shifted by one. The differential d
of this complex, which is given by counting the flow lines of V connecting critical
points with indices differ by exactly one, provides a map between the chain groups.
Then the fact that d2 = 0 implies that this is in fact a chain map.
Then, by using a similar flow line counting argument, one takes a homotopy
between two different paths connecting (f0 , g0 ) and (f1 , g1 ), and constructs a chain
homotopy between the two different chain maps constructed with the method above.
This shows that the map above does not depend on the chosen path, since the space
of all such paths is connected. What is left is to show that it is an isomorphism.
If we have paths between (f0 , g0 ) and (f1 , g1 ); and (f1 , g1 ) and (f2 , g2 ), by concatenating we get a path between (f0 , g0 ) and (f2 , g2 ). Then, we have two ways to
get a chain map between the Morse complexes of (f0 , g0 ) and (f2 , g2 ), one using
the concatenated path, and one by composing the chain maps between the Morse
complexes of (f0 , g0 ) and (f1 , g1 ); and (f1 , g1 ) and (f2 , g2 ). Again by similar methods we can construct a chain homotopy between these two. This finishes the proof,
as we can concatenate a path between (f0 , g0 ) and (f1 , g1 ) with itself in the reverse
direction, and compare the corresponding chain map with the constant path at
(f0 , g0 ).
2.2. Pseudoholomorphic Curves. Pseudoholomorphic curves were first introduced by Gromov, probably motivated by the power of the complex analytic methods. He also proved compactness results similar in nature with the Theorem 1
above. Ever since, starting with Floer, they have been used to define homology
theories similar to Morse homology for infinite dimensional spaces, which constitute a big chunk of research in Low Dimensional Topology these days. Yet, a
priori, there is no reason to expect any kind of relationship with Morse homology and pseudoholomorphic curves, except the nice behaviour of the corresponding
moduli spaces.
Even though one can make sense of a Morse function on an infinite dimensional
space, there are certain serious obstacles with defining the Morse complex in the
way described above, so one cannot really hope for a straightforward generalization.
At least so far, the infinite dimensional spaces that we really care about are the
ones that arise from finite dimensional manifolds with some extra structure. In
certain cases this extra structure gives rise to a natural almost complex structure,
and people started to really care about this after Gromov’s work. The idea of
using pseudoholomorphic curves in such cases really comes from writing down the
gradient flow equation for an appropriate metric in the infinite dimensional space,
and realizing that it looks like Cauchy-Riemann equations with respect to this
almost complex structure in certain fortunate situations. Then, also having the
4
UMUT VAROLGUNES - UMUTVG@MIT.EDU
nice compactness results in mind, one goes on and tries using pseudoholomorphic
curves in place of the gradient flow. CCH is a nice example of this procedure, as
described in [1].
2.3. Conley-Zehnder Index. The gradings of all the chain complexes that will
be constructed below are given by what is called the Maslov or Conley-Zehnder
index. We first describe Arnold’s version of the Maslov index associated to a loop
in the space of all maximal dimensional Lagrangian subspaces, Λ(n), of R2n with
the standard Riemannian, symplectic, and complex structures.
U (n) has an action on Λ(n), which induces an identification U (n)/O(n) = Λ(n).
The map det2 : U (n) → S 1 induces a well defined map Λ(n) → S 1 .Maslov index of a
loop S 1 → Λ(n) is defined to be the degree of the composition. This characterization
will be used for the grading of LCH.
The original definition later generalized a lot further in [5]. In particular, one
can give a useful definition of the Maslov index of a path in the space of lines in R2
using Remark 2.5 in [5].
The version that is used for CCH and CH is usually called Conley-Zehnder index,
and it is associated to a path in the space of symplectic matrices satisfying certain
conditions. Instead of giving the general definition, which would take a while, we
state the definition directly in the form that we will use.
Let γ be a closed nondegenerate Reeb orbit of period T in a contact manifold
(M, ξ) with a contact form α. Let us assume that in some way we symplectically
trivialized ξ along γ, where dα provides the symplectic structure on the contact
planes. Then we get a path of symplectic matrices Φt , t ∈ [0, T ] such that Φ0 = I,
and det(ΦT − I) 6= 0, where the last condition is really the definition of nondegeneracy. We call a time t crossing, if det(Φt − I) = 0. At each crossing we define a
quadratic form Γt on the kernel of Φt −I by Γt (·) = dα(·, Φ̇t ·). We can perturb Φt a
little so that all these quadratic forms are nondegenerate, and the resulting CZ index will still be well defined, because CZ index is invariant under path homotopies.
We let
X
1
sign Γt .
(6)
µCZ (γ) = sign Γ0 +
2
t crossing6=0
2.4. Stable tame isomorphism classes of DGA’s. Let A and B be two graded,
associative, unital algebras freely generated by the elements {a1 , . . . , an } and {b1 , . . . , bn }
over a commutative ring, equipped with differentials ∂A and ∂B . Let Ei denote the
special algebra which has two generators ei and ei−1 satisfying |ei | = |ei−1 | + 1 = i,
∂ei = ei−1 , and ∂ei−1 = 0.
We call a graded chain isomorphism φ : A → B elementary, if there exists
j ∈ {1, . . . , n} and u which is an element of the free subalgebra of B generated by
{b1 , . . . , bj−1 , bj+1 , . . . , bn } such that φ(ai ) = bi , for i 6= j and φ(aj ) = ±bj + u. A
composition of elementary isomorphisms is called tame.
The coproduct of A and B can be defined in a straightforward way, by just
adding up generators together, with all the other structures induced from A and
B. The degree i stabilization Si (A) of A is defined to be the coproduct of A and
Ei .
Definition 1. We call A and B stable tame isomorphic if we can stabilize A and B
finitely many times so that the resulting algebras are tame isomorphic.
HOMOLOGY THEORIES IN CONTACT GEOMETRY
5
The following purely algebraic proposition can be proved by simply showing that
stabilizing a DGA does not affect its homology groups.
Proposition 1. If A and B are stable tame isomorphic, then their homology groups
are isomorphic.
3. Cylindirical Contact Homology
Let (M, ξ) be a contact manifold with a contact form α. We pick an almost
complex structure J : ξ → ξ compatible with the symplectic form dα, and extend
it compatibly to the sympectization in the standard way. Let us also assume that
the first Chern class of ξ is trivial for simplicity.
R
Consider the action functional S on the free loop space of M given by γ 7→ γ α.
CCH is an attempt to do Floer homology in the standard way using this functional,
so we need to identify the critical points, and show that they are generically nondegenerate. The following two lemmas serve for this purpose and their proofs are
standard[10]. We always assume that α is generic in the sense of Lemma 2.
Lemma 1. γ is a critical point of S if and only if it is a closed Reeb orbit of period
S(γ).
Lemma 2. For a generic choice the contact form α the closed Reeb orbits are
nondegenerate.
Let γ be a periodic contractible Reeb orbit. We pick a spanning surface and
trivialize ξ along γ using this surface. We then define the grading of γ to be
γ = µCZ (γ) + n − 3 with respect to this trivialization. γ does not depend on the
spanning surface we choose, because of the condition c1 (ξ) = 0 [5].
Let γ m denote the m-fold cover of γ, which is itself a closed Reeb orbit. Then
for every closed Reeb orbit τ , there exists a unique embedded closed Reeb orbit γ
, and a positive integer k such that τ = γ k . We call τ a bad orbit if k is even abd
the parity of γ m is not the same for all m. If τ is not bad, then we call it good.
Definition 2. Let C∗ be the graded module freely generated by the good closed
Reeb orbits over Q
Having completed the definition of the chain complex, we now need to construct
the differential. Let Σ be a 2-sphere with a finite number of punctures, with its
standard complex structure j. Let F : (Σ, j) → (R×M, J) be a pseudoholomorphic
curve. We need to define a particular behaviour near the punctures.
Definition 3. Let γ be a closed Reeb orbit with period T . We choose polar coordinates (ρ, θ) around a puncture, and let F = (a, f ) ∈ R × M . We then say that
the puncture converges (±)’ly to γ if limρ→0 a(ρ, θ) = ±∞, and limρ→0 f (ρ, θ) =
γ(∓ θT
2π ).
We now define the moduli space that we are interested in, but in a more general
setting than what we need for CCH. In this section we will only work with s = 1.
Definition 4. Let N (γ; τ1 , . . . , τs ) denote the set of pseudoholomorphic curves F :
(S 2 \{a, b1 , . . . , bs }, j) → (R × M, J) such that a converges positively to γ, and
bi converges negatively to τi , for all i. Let us define an equivalence relation on
N (γ; τ1 , . . . , τs ), by declaring that two pseudoholomrphic curves are the same if
they differ only by a biholomorphism of (S 2 \{a, b1 , . . . , bs }, j) that is fixed at the
6
UMUT VAROLGUNES - UMUTVG@MIT.EDU
punctures. We then define the moduli space M (γ; τ1 , . . . , τs ) to be the set equivalence classes. Notice that M (γ; τ1 , . . . , τs ) has an R-action induced by translation.
The only missing piece now is an analogue of Theorem 1, which we state below
again in a more general form than what we need.
Theorem 2.PFor a generic choice of J, M (γ; τ1 , . . . , τs )/R is an orbifold of dimension γ − i τi + (n − 3)(2 − 2s) − 1.
Unfortunately, there is no way of working with Z2 coefficients in this case. So
one has to orient the moduli spaces M (γ; τ1 , . . . , τs ) in a coherent fashion [6]. In
fact, the reason that the bad orbits were excluded was to make this possible. We
omit doing this, and just assume that M (γ; τ1 , . . . , τs ) is oriented.
The differential on C∗ is then defined to be
(1)
∂(γ) = m(γ)
X
nγ,τ τ,
τ −γ=1
where m(·) denotes the multiplicity of the orbits, and nγ,τ counts the algebraic
weighted number of points in M (γ; τ )/R. We can see that the sum on the right
hand side is finite by considering the integral of the pullback of the form dα on the
punctured torus and using Stokes’ theorem, which implies that the action of such
τ has to be bounded above.
Theorem 3. If C−1 = C0 = C1 = 0, then ∂ 2 = 0 and the homology of the complex
C∗ is a contact invariant.
The proof of this theorem is really similar to the proofs of the analogous results in
finite dimensional Morse homology described above, but with many more technical
details [1], [10].
4. Contact Homology
As commented on earlier, Contact Homology is an algebraic extension of CCH.
This time instead of the free module generated by the Reeb orbits, we consider an
algebra generated by these orbits. We can extend the grading in the standard way,
and then use the whole generality of Definition 3 and Theorem 2.
Definition 5. Let A be the graded supercommutative algebra with a unit, freely
generated by the good closed Reeb orbits over Q.
The grading of a monomial in A is the sum of the grades of its factors. The
differential on the generators is defined to be
X
(1)
∂(γ) = m(γ)
nγ;τ1 ,...,τs τ1 . . . τs .
dim M (γ;τ1 ,...,τs )=1
2
Theorem 4. ∂ = 0, and the stable tame isomorphism type of A, and hence the
homology of A, is a contact invariant.
5. Legendrian contact homology
5.1. Geometric Construction. Let (M, ξ) be a contact 3-manifold with contact
form α. We will assume H1 (M ) = H2 (M ) = 0, for simplicity. When we start
working with M = R3 , this will hold automatically.
HOMOLOGY THEORIES IN CONTACT GEOMETRY
7
Let K be a Legendrian knot in M . The standard boundary homomorphism
provides an isomorphism of H2 (M, K) and H1 (K). Similar to closed Reeb orbits
above, we now consider the segments of the Reeb flow lines with both endpoints on
K. We call such segments Reeb chords. One can check that these are the critical
points of the action functional on the space of paths (I, ∂I) → (M, K). Hence they
are the natural generators of a relative Floer theory in this case. Similar to CH, we
make the following definition
Definition 6. Let A be the free associative graded unital algebra over the group
ring Z2 [H2 (M, K)] = Z2 [H1 (K)] generated by the Reeb chords.
The gradings are given by the following procedure. For each Reeb chord a(t), we
fix a capping path γa inside K, which connects the two endpoints of a(t). Then, we
pick a surface that spans a ∪ γa and, symplectically trivialize ξ along a ∪ γa . This
gives us a path of Lagrangian subspaces of R2 , by considering the subbundle of
ξ |a∪γa given by Tγa (t) K along γa , and DΦ(t) · Ta(0) K along a, where Φ is the Reeb
flow. The Maslov index of this path is defined to be the grading of a. Note that
this path does not close up so one either has to close up the path in a canonical way
(this is done in [11]), or use the more general definition given in [5]. The grading
of the extra generator of the algebra A, the generator of H1 (K), is given by twice
the rotation number of K.
Let us fix an almost complex structure J on the symplectization (M ×R, d(et α)).
We now define the differential of A.. We will consider pseudoholomorphic curves
F : (D∗2 , ∂D∗2 ) → M × R, where D∗2 is a disc with finitely many punctures on its
boundary.
Definition 7. Let a be a Reeb chord. A neighborhood of a puncture in D∗2 is
conformally equivalent to (0, ∞) × [0, 1] with coordinates (s, t) such that s → ∞ is
equivalent to approaching the puncture. Let F = (r, f ) ∈ R×M . Then, we say that
the puncture converges (±)’ly to a if lims→∞ r(s, t) = ±∞, and lims→∞ f (s, t) =
a(t).
Definition 8. Let N c (x; y1 , . . . , yn ) denote the set of pseudoholomorphic curves
F : (D∗2 , ∂D∗2 ) → M × R, where the punctures are denoted by a, b1 , . . . , bn , such
that a converges positively to x, and bi converges negatively to yi , for all i, with
the extra requirements,
R
(1) D2 F ∗ dα < ∞
∗
(2) f (∂D∗2 ) ∪ γ1 ∪ −γb1 ∪ . . . ∪ −γbn . represents the homology class c ∈ H1 (K).
Let us define an equivalence relation on N c (x; y1 , . . . , yn ) by declaring that two
pseudoholomrphic curves are the same if they differ only by a biholomorphism of D∗2
that is fixed at the punctures. We then define the moduli space M c (x; y1 , . . . , yn )
to be the set equivalence classes. Notice that M c (x; y1 , . . . , yn ) has an R-action
induced by translation.
P
Theorem 5. dim M c (x; y1 , . . . , yn ) = |x| − |yi | + 2r(K) · c.
We can now define the differential in the usual way. Let us denote the generator
of H1 (K) by t. Then,
X
(1)
∂(x) =
(#M/R)tc y1 . . . yn ,
dim M c (x;y1 ,...,yn )=1
where # counts the mod 2 number of elements of the moduli space, and ∂(t) = 0.
8
UMUT VAROLGUNES - UMUTVG@MIT.EDU
5.2. Combinatorial Construction. From now on we only work with R3 with
its standard tight contact structure ξst , and contact form dz + xdy. We want to
translate the geometric theory above to a combinatorial one in this case. Let us
also fix a Legendrian knot K in R3 . We will mostly work with the Lagrangian
projection of K.
The first observation is that the Reeb vector field is simply the vector field ∂z .
Hence, the Reeb chords correspond bijectively to the crossings in the Lagrangian
projection of the knot K.
Definition 9. Let a1 , . . . , an be the crossings in the Lagrangian projection of K.
Then A is defined to be the graded free associative unital algebra over Z2 [t, t−1 ]
generated by a1 , . . . , an .
The grading of t is defined to be 2r(K), as expected. To define the gradings
of the generators we choose capping paths γai . This is the path that starts at the
undercrossing of ai and ends when it comes back to ai for the first time again. Let us
perturb the diagram a little, so that at the crossings two strands are perpendicular
to each other. This lets us define the fractional rotation number of the capping
paths r(γai ), which takes values in odd multiples of 41 . Then the grading of a is
defined to be −2r(γai ) − 12 .
Let us also define the Reeb signs of a crossing in the following way. We can
approach to the crossing from the under strand in the two possible ways. We label
the quadrants that stay on our left just before we get to the crossing (+), and the
ones that stay on our right (-). This gives (+) or (-) labels to each quadrant around
each crossing. This convention is related to the fact that the pseudoholomorphic
curves can converge to the Reeb chords, either positively or negatively, and will be
used to make that distinction soon. It is not related to orientations.
We will now define the differential by counting immersed discs in the xy-plane,
which will be denoted by R2 .
Definition 10. Let ∆c (a; b1 , . . . , bn ) be the space of all orientation preserving immersions g : (D∗2 , ∂D∗2 ) → (R2 , K) modulo reparametrizations of the domain, satisfying
the following conditions,
(1) (πxy |K )−1 (Im(f |∂D2 )) ∪ γ1 ∪ −γb1 ∪ . . . ∪ −γbn represents the homology
class c ∈ H1 (K)
(2) g sends the boundary punctures to crossings of the diagram. At a crossing
that is the image of a puncture, the image of g covers one or three crossings,
and at a the majority of crossings is positive, whereas at bi it is negative.
We then define the formal dimensions of these spaces by the familiar formula,
X
(2)
dim ∆c (a; b1 , . . . , bn ) = |a| −
|bi | + 2r(K) · c − 1.
The following lemma is a straightforward consequence of this definition.
Lemma 3. The elements of ∆c (a; b1 , . . . , bn ) have exactly dim ∆c (a; b1 , . . . , bn )
non-convex corners
We now define the differential as follows.
X
(3)
∂(a) =
(#∆)tc b1 . . . bn .
dim ∆c (a;b1 ,...,bn )=0
Lemma 3 implies that we only count the immersions with no nonconvex corners.
HOMOLOGY THEORIES IN CONTACT GEOMETRY
9
Theorem 6. ∂ 2 = 0, and the stable tame isomorphism type of A is a Legendrian
isotopy invariant of the knot K
The proof of ∂ 2 = 0 can really be summarized with following diagram, which
provides a pairing between the nonzero terms in the expansion of ∂ 2 with respect
to the standard basis.
The proof of the invariance relies strongly on the Reidemeister moves for Legendrian isotopy. Namely, one relates the algebras corresponding to two diagrams
which differ by only one Reidemeister move using stablizations. Here, we see the
importance of having defined the more general equivalence relation of stable tame
isomorphism.
5.3. Equivalence of the Constructions. Let us first note that the following
proof does not prove that the geometric construction is well defined, as it depends
strongly on particular choices of a contact form and an almost complex structure.
The invariance under these choices is proved in [11]. Hence, what we will be proving
is the following: if we let the contact form α to be dz +xdy, and the almost complex
structure J to be chosen to satisfy J(∂x ) = ∂y − x∂z , and J(∂y ) = −x∂t − ∂x in
addition to the other properties that it is required to satisfy in general, then, with
the obvious notation, Acombinatorial and Ageometric are isomorphic as differential
graded algebras. We consider the obvious map Acombinatorial → Ageometric , that
sends a crossing to the corresponding Reeb chord. It suffices to check that this map
preserves the gradings, and the differential.
Grading: Here we note one more technical issue. Recall that in the geometric
picture, the choice of capping paths were made explicit in the construction, but
were arbitrary, whereas in the combinatorial picture we had certain choices for the
capping paths. We need to choose the same capping paths in the geometric picture
as the ones in the combinatorial picture. If we want a theory which is independant
of the choice of the capping paths, we lose the Z grading, and get a Z2r(K) graded
algebra. See [11] for a discussion of this approach.
Notice that we can symplectically trivilize the contact planes globally by the
vector fields ∂x and ∂y − x∂z . Since the Maslov index of a path is independant
of the trivialization we choose [5], we can use this trivialization to grade the Reeb
chords. Then using the alternative characterization given in Section 2.3, one can
see that computing the Maslov index in this setting is the same as computing the
rotation number, up to a factor of two, and a correction in the end.
Differential: We prove the following theorem, which trivially implies that the
coefficients in the basis expansion of the differential are the same. This is because
if either moduli spaces are zero dimensional, they are finite.
Theorem 7. There exists a bijection between M c (x; y1 , . . . , yn )/R and ∆c (x; y1 , . . . , yn ).
The key step is to define the intermediate moduli spaces ∆ch (x; y1 , . . . , yn ). We
first notice that the projections of maps in M c (x; y1 , . . . , yn ) to the xy-plane are
holomorphic maps with respect to the standard coordinate structure of C, i.e they
satisfy the Cauchy-Riemann equations with respect to the coordinates x, y. This is
shown by direct computation.
This leads to the following definition.
Definition 11. Let ∆ch (a; b1 , . . . , bn ) be the space of all holomorphisms g : (D∗2 , ∂D∗2 ) →
(R2 , K) modulo biholomorphic reparametrizations of the domain, satisfying the following conditions,
10
UMUT VAROLGUNES - UMUTVG@MIT.EDU
(1) (πxy |K )−1 (Im(f |∂D2 )) ∪ γ1 ∪ −γb1 ∪ . . . ∪ −γbn represents the homology
class c ∈ H1 (K)
(2) g sends the boundary punctures to crossings of the diagram. At a crossing
the image of g covers one or three crossings, and at a the majority of
crossings is positive, whereas at bi it is negative.
We have the obvious maps ∆ch (a; b1 , . . . , bn ) → ∆c (a; b1 , . . . , bn ),
and M c (a; b1 , . . . , bn )/R → ∆ch (a; b1 , . . . , bn ), where the first map is just the map
induced by the inclusion of the holomorphic maps into the orientation preserving immersions, and the second one is the map induced by M c (a; b1 , . . . , bn ) →
∆ch (a; b1 , . . . , bn ), projecting the pseudoholomorphic maps to the xy plane. The
fact that the second map is well defined follows from the following claim.
Claim 1. Let f ∈ M c (a; b1 , . . . , bn ) then the asymptotic behaviour of f on the
boundary punctures imply that πxy ◦ f satisfies the second condition of Definition
2.
We consider the image of the right part of the boundary of the strip model
around the puncture after the projection. As we approach infinity along this line,
we approach to the crossing in the Lagrangian projection. It suffices to determine
from which strand we approach the crossing, and before we get to the crossing
which direction the image of the outward pointing normal vector is pointing to. If
the puncture converges positively, we find that the image is the lower strand, and
if negatively, it is the upper one. Using that the map is holomorphic finishes the
analysis.
We now show that these maps are both bijections, which finishes the proof.
Claim 2. Every equivalence class in ∆c (a; b1 , . . . , bn ) contains a holomorphic representative.
This claim follows directly from the Riemann mapping theorem. Hence, the map
∆ch (a; b1 , . . . , bn ) → ∆c (a; b1 , . . . , bn ) is bijective.
Claim 3. If we are given f ∈ ∆ch (a; b1 , . . . , bn ), f = (x(u, v), y(u, v)), we can lift f
uniquely to p(f ) ∈ M c (a; b1 , . . . , bn ) such that p(f ) = (x(u, v), y(u, v), z(u.v), t(u, v)),
up to translation in the t coordinate.
The Cauchy Riemann equations for a map D2 → R3 ×R to be pseudoholomorphic
are the following.
(4)
∂u x − ∂v y = 0 and ∂u y + ∂v x = 0
(5)
∂u t − ∂v z = x∂v y and ∂u z + ∂v t = x∂u x.
We first show that f can be lifted uniquely (up to a constant) to p(f ) such that
these are satisfied.
Using (5), we conclude that z(u, v) has to be harmonic. By using that the image
of the boundary of the disc is a Legendrian knot, we can recover what z has to be
on the boundary. Then, by the existence and the uniqueness of the solutions for
the Dirichlet problem we prove the uniqueness of z.
We claim that for this particular choice of z there exists a unique t(u, v) that
solves (5), up to a constant. Indeed, by simple computation (∂v +x∂v y)du+(x∂u x−
∂u z)dv is an exact differential. Hence, Poincaré Lemma does the job.
HOMOLOGY THEORIES IN CONTACT GEOMETRY
11
Now, we are only left with showing that p(f ) satisfies the conditions of Definition
8. The homology condition is trivially satisfied. The asymptotic conditions at
infinities are checked by a local computation. Namely, one explicitly constructs a
model lift of the holomorphic disc around a puncture. It is not hard to show that
the asymptotic conditions are satisfied for this particular lift. A theorem of Robbin
and Salamon combined with the maximum principle for harmonic functions show
that an arbitrary lift has to be exponentially close to this model, as we get close
to the puncture [4], [12]. Hence, we get the desired convergence results from the
model computation.
The finite energy condition can also be checked using this model lift. By Stokes’
Theorem the energy is equal to the integral of the contact form along the image of
the boundary of the punctured disc. This can also be seen as an integral over the
boundary of the disc, by change of variables. This integral is concentrated on the
parts of the boundary that are close to a puncture. We then cut out small opens
discs around each puncture, and consider the integral of the same form along the
boundary of the remaining part. If the discs are small enough we can estimate the
integral along the boundaries of the discs using the model lifts, and the rest of the
integral is small compared to this contribution. This finishes the proof by observing
that the integral has finite limit, as the radii of the discs go to zero.
References
[1] Ya. Eliashberg, A. Givental, and H. Hofer. An introduction to symplectic field theory, Geom.
Funct. Anal., 2000, Special Volume, Part II, 560673.
[2] Y. Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441483.
[3] Y. Eliashberg, Invariants in contact topology, Proceedings of the International Congress of
Mathematicians, Berlin, 1998, Doc. Math. Extra vol. II (1998) 327338 (electronic).
[4] J. B. Etnyre, L. L. Ng and J. M. Sabloff, Invariants of Legendrian knots and coherent orientations, J. Symplectic Geom. 1 (2002) 321367.
[5] J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827844.
[6] A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Zeit., 212 (1993), 1338.
[7] M. Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations. J. Eur. Math. Soc. (JEMS) 4 (2002), no. 4, 313361.
[8] C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology I, Geom.
Topol. 14 (2010) 24972581.
[9] M. Hutchings, Lecture notes on Morse homology (with an eye towards Floer theory and
pseudoholomorphic curves), Fall 2002, Berkeley.
[10] F. Bourgeois, Introduction to Contact Homology, June 2003, Summer School in Berder.
[11] T. Ekholm, J. Etnyre, M. Sullivan, The contact homology of Legendrian submanifolds in
R2n+1 , J. Differential Geom. 71 (2005) 177305.
[12] J. W. Robbin, D. A. Salamon, Asymptotic behaviour of holomorphic strips, Ann. Inst. H.
Poincar Anal. Non Linaire 18 (2001), no. 5, 573612.