Mathematics Department
Master Thesis
Branched Covers, Open Books
and Contact Structures
Advisor:
Prof. Paolo Lisca
Eric Stenhede
Candidate:
Academic Year 2019/2020
12 June 2020
In this thesis we study the relationship between branched covers, open book decompositions and contact structures in dimension three.
The main result shown is the one to one correspondence between contact structures and open book decompositions up to positive stabi-lization, now known as “Giroux’s correspondence”.
Then, we proceed to define contact branched covers and, using Giroux’s correspondence together with a theorem by Montesinos and Morton, we show that every closed contact 3-manifold can be obtained as a 3-fold simple branched cover with base manifold the standard contact 3-sphere (S3, ξ
std) and branch set a transverse link.
Finally, we present a recent result by Etnyre and Casals concerning the existence of a 4-components transverse universal link. This is a transverse link in the standard contact 3–sphere such that all closed contact 3–manifolds are contact branched covers over this link.
Introduction 2
1 Branched Covers and Open Books 3
1.1 Ordinary Covering Spaces . . . 3
1.1.1 The Monodromy . . . 4
1.2 Branched Covers of Manifolds . . . 4
1.2.1 Surfaces . . . 4
1.2.2 3-Manifolds . . . 6
1.3 Universal Links and Knots . . . 7
1.3.1 Monodromy Representations . . . 8
1.3.2 Admissible Transformations . . . 10
1.4 Open Book Decompositions . . . 12
1.5 Covers of Open Book Decompositions . . . 19
2 Contact Topology 21 2.1 Contact Manifolds . . . 21
2.1.1 Examples of Contact Manifolds . . . 21
2.1.2 Gray Stability Theorem . . . 26
2.1.3 Darboux’s Theorem and Neighborhoods Theorems. . . 26
2.1.4 Isotopy Extension Theorems . . . 29
2.2 Tight and Overtwisted . . . 30
2.3 Knots in Contact 3-Manifolds . . . 31
2.3.1 Legendrian and Transvese Knots . . . 32
2.3.2 Front Projection . . . 32
2.3.3 Approximation Theorem . . . 34
2.3.4 Classical invariants . . . 35
2.3.5 Stabilizations . . . 36
2.4 Convex Surfaces . . . 37
2.5 Gluing Tight Contact Manifolds . . . 40
3 Open Books and Contact Structures 43 3.1 Orientations . . . 44
3.1.1 Contact Structures . . . 44
3.1.2 Open Books . . . 44
3.2 Stabilization of Open Books . . . 45
3.3 Compatibility . . . 47
3.4 Giroux’s Correspondence . . . 50
3.4.1 From Open Books to Contact Structures . . . 50
3.5 Branched Covers and Contact Structures . . . 63
4 A Transverse Universal Link 67 4.1 Statement of the Main Theorem . . . 67
4.2 The Front Projection in T2×(0, 1) . . . 69
4.3 Admissible Contact Transformations . . . 70
Contact geometry, as a subject in its own right, was born in 1896 in the monumen-tal work of Sophus Lie on Berührungstransformationen (contact transformations). However, contact geometric notions can be find in the work of Christiaan Huygens on geometric optics in the Traité de la Lumière of 1690. In the last thirty years, this field has undergone an exponential growth and many exciting results continue to appear nowadays, specially in high dimensions.
In this thesis we hope to give the reader an insight of this beautiful subject, even if we deal with a small part of it, restricting ourself to the three dimensional setting and investigating contact manifolds from a topological point of view. The ideal reader should have some basic knowledge on smooth manifolds, branched covers and contact topology, say on the level of the textbooks by Lee [38, Chapters 1-15], Prasolov–Sossinsky [47, Chapter 7] and Geiges [24, Chapters 1-3].
A contact structure ξ is a maximally non-integrable plane field on a 3-dimensional manifold M, meaning that it is impossible to find a surface S, even locally, such that the tangent space T S equals ξ|S. It turns out that there are no local invariants for
contact structures, in contrast with Riemannian geometry, for instance, where the local structure coming from the curvature gives rise to a rich theory. The interesting questions in contact geometry thus appear only at the global level. However, it is actually the local flexibility that allows us to use purely topological tools such as branched covers or open book decompositions to prove strong global theorems, such as the existence of contact structures in any orientable closed 3-manifolds.
At the beginning of the first Chapter we give the definition of branched covers and we recall some important theorems about them. Essentially, a branched cover is a map between manifolds such that, away from a set of codimension two (called the
branch locus), it is a honest covering map. Starting from branched covers between
surfaces, we then study branched covers between 3-manifolds giving particular atten-tion to these that have the sphere S3 as target space. Note that, in dimension three,
the branch locus can be either a link or a knot, so, in particular, we will see an im-portant theorem, due to Alexander, stating that every closed orientable 3-manifold admits a branched covers with target space the sphere S3 and branch locus a link.
Moreover we give the definition of universal links that are links with the property that every closed orientable 3-manifold can be obtained as a branched cover with branch locus this links. Another important theorem due to Hilden and Montesinos state the existence of an universal knot. We then proceed to define open book de-compositions and proving their existence in every closed orientable 3-manifold. At the end of the chapter we present a theorem by Montesinos and Morton explaining the relation between branched covers and open book decompositions.
In the second Chapter we introduce contact structures, giving many examples and fundamental theorems on the subject. In particular we see that many results
coming from ordinary smooth topology carries over to the contact setting. We then study surfaces inside contact manifolds, giving particular attention to those, called convex surfaces, which have a special kind of tubular neighborhood where the contact structure is vertically invariant. Finally we study transverse and Legendrian knots, that are, respectively, knots everywhere transverse or tangent to the contact planes.
With Chapter 3 we begin the main part of the thesis. In this chapter we present “Giroux’s correspondence” that is the one to one correspondence existing between contact structures and open book decompositions up to positive stabilization. At the end of the chapter we define contact branched covers and prove a strengthening of Alexander Theorem for contact manifolds.
In Chapter 4 we present a contact version, due to Etnyre and Casals, of the Hilden-Montesinos Theorem. We prove, indeed, that there exists a transverse (uni-versal) link in the standard contact structures on the 3–sphere such that all contact 3–manifolds are contact branched covers over this transverse link.
Branched Covers and Open Books
In this chapter we recall some basic facts and definitions about branched covers and open book decompositions and how they can be used to present 3-manifolds. These concepts will be used throughout all the thesis.
We will work exclusively, except when specified, in the smooth category, this is to say that all the maps and manifolds involved are supposed to be smooth. However most of the theorems presented in this chapter have only been proven in the topological case but it is clear, if one looks trough the proofs, that they hold in the smooth category as well.
In the first sections we shall see that any closed orientable 3-manifold can be presented as a finite covering of the sphere S3 branched along a knot or a link. In
fact there exist certain universal knots and links in S3 (e.g. the Borromeo rings)
from which any 3-manifold can be obtained as the total space of a covering whose branching set is that specific link. Then we introduce the definition of open book decomposition and prove that every closed orientable 3-manifold admits an open book decomposition. This will be done using branched covers. Finally, as a sort of converse, we present (and give a sketch of a proof of) of an important theorem by Montesinos and Morton that shows how every open book decomposition gives rise to a branched cover.
Almost all the material about branched covers is taken from [47], while the part regarding open book decomposition is mostly taken from [13].
1.1
Ordinary Covering Spaces
Definition 1.1.1. A continuous map p : M → N between topological spaces is
called a covering if there exists an open cover {Uα} of N such that for each α,
p−1(Uα) is a disjoint union of open sets in M, each of which is mapped
homeomor-phically onto Uα by p.
It will be helpful to review some facts from algebraic topology about covering spaces. First, we recall an important classification theorem for covering spaces.
Theorem 1.1.2. Let X be a CW-complex. The isomorphism classes of connected
coverings of X preserving base points are in 1-1 correspondence with the subgroups of π1(X, x0).
Remark 1.1.3. This relationship is given by associating to any covering space p :
1.1.1
The Monodromy
Given a connected n-fold covering space p : ˜X → X, we get a homomorphism m: π1(X, x0) → Sn(where Snis the symmetric group over n symbols) as follows: let
x1, ..., xn be any fixed numbering of the points in p−1(x0). Given a loop γ : S1 → X
based at x0, let γi be the lift of γ to a path beginning at xi. The other end point
of the path will be a point xk. We define σγ(i) = k. Clearly σγ is an element of
Sn and one can easily check that it is independent of the homotopy class of γ as a
based loop. Thus we can define m([γ]) = σγ where [γ] is the element of π1(X, x0)
that γ defines. Notice that if we ordered the points in another way then we would get a homomorphism that is conjugate to the one above.
So, for every connected n-fold covering space we get a conjugacy class of repre-sentations called the monodromy of the covering space. Notice that if the covering space is not connected we still get a monodromy representation.
Lemma 1.1.4. If p: ˜X → X is an n-fold covering space then ˜X is connected if and only if the image of the monodromy acts transitively on {1, ..., n}. More precisely the number of components of ˜X is precisely the number of orbits in {1, ..., n} of the action of the image of the monodromy.
Given a connected manifold X and a homomorphism m : π1(X, x0) → Sn, choose
one representative i1, ..., ik from each orbit in {1, ..., n} of the action of π1(X, x0).
Let Hj = {g ∈ π1(X, x0) : m(g)(ij) = ij} and ˜Xj the covering space corresponding
to Hj. If ˜X = ∪kj=1X˜j then ˜X → X is a covering space of X for some labeling of
the points p−1(x
0) one may check that the monodromy of p is m.
Example 1.1.5.Consider a covering p : X → S1. The group π
1(S1, x0) is generated
by one element, call it γ. Let the image of γ under the monodromy be (146)(23)(5) ∈
S6. Then, the space X consists of three disjoint copies of S1. This three copies of
S1 give respectively a 3-fold, a 2-fold, and a 1-fold cover of S1.
1.2
Branched Covers of Manifolds
For most of this chapter we will be interested in branched covers. Essentially, a branched cover is a map between manifolds such that away from a set of codimension 2 (called the branch locus) p is a honest covering. The material presented here is mostly taken from [47].
1.2.1
Surfaces
Definition 1.2.1. A smooth map p : F → S between 2-dimensional smooth
man-ifolds is said to be a branched cover if every x ∈ S has a neighbourhood U with the following property. For each component U0 of p−1(U) there is a number k ∈ N
and a commutative diagram
D2 U0
D2 U
h0
pk pU
where h0 and h are diffeomorphisms, h0(0) = x, and with D2 regarded as a subset
of C the map pk is described by
pk(z) = zk.
The natural number k is called the branching index of the point p−1(x)∩U. Write
C ⊂ F for the 0-dimensional submanifold made up of points with branching index
greater than 1. The points p(C) ⊂ S are called the branch points. Moreover a branched cover is simple if every every point in F has branching index 1 or 2.
In other words if we have a branched cover p : F → S and we restrict it to the complement of the counter image of the branch points we get a covering. Moreover It is easy to see that for an n-fold branched cover, the sum of branching indices of all the preimages of any branch point is equal to n.
As the definition already suggest the simplest example of branched cover is the following. The example describes for different n the structure of an arbitrary branched covering near its branch points.
Example 1.2.2. Let D2 = {z ∈ C : |z| ≤ 1} and let p : D2 → D2 be the map
given by the formula p(z) = zn. Then p is an n-fold branched covering with unique
branch point z = 0.
Definition 1.2.3. Suppose ˜p : F → S is a branched cover between surfaces. We
shall say that a diffeomorphism g : F → F covers a given diffeomorphism f : S → S with respect to ˜p if ˜p ◦ g = f ◦ ˜p.
The following proposition shows that every compact orientable surface is a branched cover over the sphere.
Proposition 1.2.4. Let Sg be the orientable closed surface of genus g. Then, there
exists a 2-fold branched covering p : Sg → S2.
If we give up the requirement on the order of the covering, we can specify the number of branch points.
Theorem 1.2.5. Let Sg be the orientable closed surface of genus g ≥1. Then there
exists a branched cover p: Sg → S2 with exactly three branch points.
The Riemann-Hurwitz formula
Recall that if Sg,dis an orientable surface with genus g and d boundary components,
then the Euler characteristic of Sg,d is given by the formula
χ(Sg,d) = 2 − 2g − d.
The Euler characteristic is a tool for idenifying a surface. Recall that any orientable surface is determined up to diffeomorphism by its Euler characteristic and the num-ber of its boundary components. For an n-fold covering map p : F → S, we have the relationship χ(F ) = nχ(S). The Riemann-Hurwitz formula generalizes this to the case of branched covers.
Theorem 1.2.6 (Riemann-Hurwitz formula). Suppose that p : F → S is an n-fold
branched covering of compact 2-manifolds, y1, ..., yl are the preimages of the branch
points, and d1, ..., dl are the corresponding branching index. Then,
χ(F ) +
l
X
i=1
(di−1) = nχ(S).
The Riemann-Hurwitz formula has numerous applications. The first one that we discuss has to do with Theorem 1.2.5, which asserts the existence of a branched covering p : Sg → S2 with exactly 3 branch points (when g ≥ 1). Can this number
be decreased? The answer is “no”, as the next easy proposition shows.
Proposition 1.2.7. If g ≥1, there exists no branched covering p : Sg → S2 of the
sphere by a genus g surface having less than3 branch points.
1.2.2
3-Manifolds
Many definitions and theorems from the previous section can be carried over to the case of dimension 3 without significant modifications.
Definition 1.2.8. A smooth map p : M → N between 3-dimensional smooth
manifolds is said to be a branched cover if every x ∈ N has a neighbourhood U with the following property. For each component U0 of p−1(U) there is a number
k ∈ N and a commutative diagram
D2×[−1, 1] U0
D2×[−1, 1] U
h0
pk pU
h
where h0 and h are diffeomorphisms, h0(0) = x, and with D2 regarded as a subset
of C the map pk is described by
pk(z, t) = (zk, t).
In this situation M is called the covering manifold, N is the base and the natural number k is called the branching index of the point p−1(x) ∩ U. Write C ⊂ M
for the 1-dimensional submanifold made up of points with branching index greater than 1. The submanifold p(C) ⊂ N is the branch set. Moreover a branched cover is simple if every every point in M has branching index 1 or 2.
Remark 1.2.9. The branch set L is sometimes called the downstairs branch set.
There is some confusion in the literature, however, about what to call the upstairs branch set. Some authors use this term to refer to C, others use it for the set p−1(L).
As we will see, C is, in general, a strict subset of p−1(L), i.e. there are points in the
preimage of the branch set where the map p is unbranched.
Notice that both the maps p|M \p−1(L) : M \ p−1(L) → N \ L and p|C : C → L
are unbranched coverings. Likewise, p is an unbranched covering when restricted to a neighbourhood of the 1-dimensional submanifold p−1(L) \ C.
We have seen that every orientable closed surface is a branched cover of S2. The
same holds in the 3-dimensional setting. The first result in this direction have been given by Alexander in 1920.
Theorem 1.2.10 (Alexander Branched Cover Theorem. Alexander 1920, [2]).
Ev-ery closed orientable smooth3-manifold is a branched cover of S3 with branch set a
link.
This result shows that branched covers are not simply a method for constructing some 3-manifolds, but a tool for constructing every 3-manifold. Yet this is simply an existence result; the degree of the covers could be arbitrarily large and the links could be very complicated. One would like to know if, as with surfaces, restrictions can be placed either on the branch locus or on the number of folds and still construct every 3-manifold. The following theorem greatly improves the result by Alexander.
Theorem 1.2.11(Hilden 1974, [27]; Montesinos 1976, [42]). Every closed orientable
3-manifold is a 3-fold simple branched cover of S3 with branched set a knot.
Note that this statement strengthens the Alexander theorem in three directions: 1. it specifies the number of folds of p;
2. gives an upper bound for the branching index; 3. the branching set may be assumed connected.
Thus the Hilden-Montesinos theorem provides a bridge between knot theory and the theory of 3-manifolds: the latter are none other than 3-fold simple branched covers of the 3-sphere branching along different knots.
1.3
Universal Links and Knots
In the previous section we saw that all surfaces can be constructed by either looking only at covers with three branched points or looking only at 2-fold cyclic covers. Hilden and Montessinos showed that we can look only at all 3-fold covers to obtain all 3-manifolds. Could we also look only at covers over one fixed branch locus, or over a finite set of knots and still construct all closed orientable 3-manifolds, mirroring the result for surfaces? The following theorems show that this is indeed the case.
Definition 1.3.1. A link L ⊆ S3 is said to be universal if any closed orientable
3–manifold can be obtained as a branched cover of S3 along L.
Theorem 1.3.2(Thurston 1982). There exists a 6-component universal link in the
3–sphere.
Unfortunately, since the result has never been published, we couldn’t find a picture of this link. However, later it was discovered that also the Whitehead link and Borromean rings are universal.
Theorem 1.3.3 (H. Hilden, M. Lozano, and J.M. Montesinos 1983, [28]). The knot
Figure 1.1: The universal knot found by Hilden, Lozano and Montesinos. In the following subsections we discuss some techniques for defining branched covers of the sphere S3 by means of links represented by diagrams whose arcs are
assigned “colors”, i.e., marked in a certain way by elements of the permutation group. These techniques have been used to produce the universal knot of Theorem1.3.3and we are interested in them primarily because they will be needed in Chapter 4, where we strengthen the result by Thurston. To attain this result, we need modifications of the branch set that do not change the covering manifold. Such modifications are provided by colored link diagram techniques.
1.3.1
Monodromy Representations
As we already saw at the beginning of the chapter, a covering map p : M → Y , for M connected, is determined by its monodromy representation. Let x0 ∈ Y be a
fixed base point and label the points in p−1(x
0) = {x1, ..., xn}. Then, given a loop γ
in Y based at x0 we can define the element σγ of the symmetric group Sn by σγ(i)
representation of the fundamental group
r: π1(Y, x0) → Sn= Aut(p−1(x0)).
Moreover, given such a representation the subgroup p∗π1(M, x1) ⊆ π1(Y, x0) gives a
covering space M that corresponds to the representation.
Let p : M → S3 be a branched cover map with branch locus L ⊂ S3. By
definition, the restriction p|(M \p−1(L)) : M \ p−1(L) → S3 \ L is a covering space
and, by the above discussion, this restriction is determined by a representation
π1(S3 \ L, x0) → Sn, with n = |p−1(x0)| the cardinality of the fiber over any x0 ∈
S3 \ L. In turn, such a representation determines the initial branched cover map
p : M → S3. Indeed, the representation of the fundamental group determines a
cover of S3\ L and, in addition, any covering of S3\ Lcan be uniquely extended to
a branched covering map as follows. Identify a neighborhood Ui of each component
Li of L with Li × C with coordinate (θ, z) with θ ∈ Li ∼= S1. If Ui is chosen
sufficiently small then, the counter image of this neighborhood by the cover of S3\ L
given by a representation of the fundamental group is a disjoint union of open sets diffeomorphic to S1×
C \ {0}
and the cover is given in this neighborhood by
S1×C \ {0}→ L ×C \ {0}, (θ, z) 7→ (mθ, zn).
It is thus clear that, after we glue copies of S1× {0} on the manifold upstairs, there
is a well define branch cover from a new compact manifold M to S3 with branch set
L.
The fundamental group π1(S3\L, x0) is well-known to be generated by loops as in
Figure1.2. In particular, given a diagram for L we have the Wirtinger presentation for π1(Y \ L, x0) with generators xi in 1-1 correspondence with the strands in the
diagram and relations coming from the crossings: xi = x−1k xjxk, respectively xi =
xkxjx−1k , at a right handed, respectively left handed, crossing, where strand k goes
over incoming strand j and outgoing strand i.
x0
Figure 1.2: Generators of π1(S3\ L).
Definition 1.3.4. A diagram of a link L where the strands are labeled with elements
of Snthat satisfy the required relations at the crossings is called a colored diagram
or colored link.
Thus a colored link L determines a representation of πi(S3\L, x0) and so a unique
branch cover. It is easy to check whether a branched cover given by a colored link is simple: one just needs the strands to be labeled by transpositions, as higher length permutations correspond to more than two points coming together.
1.3.2
Admissible Transformations
Consider a transformation of a colored diagram that produce a new colored diagram in such a way that the upstairs manifolds of the branched covers given by these colored links are diffeomorphic. Such a transformation of a colored diagram will be called admissible.
For example, plane isotopies and Reidemeister moves are admissible transforma-tions. In fact every Reidemester move comes from a diffeomorphism h : S3 → S3
isotopic to the identity so if p : M → S3 is a branched cover, then h ◦ p : M → S3
is a branched cover whose branch set diagram (using the same projection as for the original diagram) is obtained from the branch set diagram of p by plane isotopies and Reidemeister moves. The following very useful lemma gives other admissible transformations.
Lemma 1.3.5. Let L be a link in a 3–manifold Y and p : M → Y be a branched
covering map with branch set L. If part of a diagram for L is as shown on one side of a row in Figure 1.3, then replacing that portion of L with the other diagram shown in that row will result in a new branched covering of Y that still yields the same manifold M . In the third and fourth transformation we ask i, j, k to be all distinct. (i j) (j k) (i j) (j k) (i j) (j k) (i k) (i j) (j k) (i j) (i j) (j k) (i j) (i j) (j k) (i j) (i j) (j k) (i k) (i k) (j k) (i j) (i j) (j k) (i j) (i j) (k l) (k l) (i j) (k l) (k l) (i j) (i j) (j k) (i j) (j k) (i j) (j k) (i k) (i j) (j k)
Figure 1.3: Replacing a portion of the branch set of a simple cover with one of the figures in a row with the other does not change the manifold described by the branched cover.
In order to prove this lemma we first need to do a little detour. We give an example of a branched cover determined by coloring its branch locus. For this purpose note that the same considerations on coloring the branch locus hold if we consider branched covers between surfaces with base manifold the disk D2. Indeed,
the situation in this case is even simpler: any coloring of a finite set of points in the interior of D2 gives rise to a branched cover. Figure1.4 shows an example.
Moreover if f : D2 → D2 is the composition of three half-Dehn twist (each of
(12) (23)
˜p 1 2 3
Figure 1.4: A 3-fold simple branched cover D2 → D2.
1.4, then the diffeomorphism g of Figure1.5 covers f. For futures applications note that g and f can be chosen to be the identity on the neighborhood of the boundary.
˜p
g f
˜p
Figure 1.5: The first row describe the branched cover while the columns show the diffeomorphisms g and f.
Now suppose the base of a branched cover p : M → N is obtained by gluing together the manifolds N1and N2, while the cover is obtained by gluing the manifolds
M1 and M2 (not necessarily connected), and Mi = p−1(Ni). We can assume that
∂N1 = ∂N2 = S and ∂M1 = ∂M2 = F , where the gluings are performed along the
identity of the surfaces F and S.
Lemma 1.3.6. Suppose ˜p is the restriction of the map p to the surface F . Assume
to N2 and M1 to M2 along the diffeomorphisms f : ∂N1 → ∂N2 and g: ∂M1 → ∂M2
respectively, the map p remains a branched cover if and only if g covers f .
Proof. The condition ˜p◦g = f ◦ ˜p is necessary and sufficient for the map p to remain
well-defined and continuous after the identifications described in the lemma are performed. The fact that the manifold S intersects the branching set transversally implies that the map ˜p : F → S is a branched covering (of surfaces). Since f and
g are homeomorphisms satisfying ˜p ◦ g = f ◦ ˜p, it follows that g maps p−1(f−1(x))
bjectively onto p−1(x). Hence f takes branch points to branch points, while g takes
the preimage of each branch point to the preimage of a branch point. Thus in our identifications we actually perform identifications of the branch set and its preimage. Therefore after all these identifications, the restriction of p to the complement of the branch set will still be a covering.
Proof of Lemma1.3.5. The transformation on the first two rows are just isotopies
and Reidemeister moves that we already know are admissible.
For the transformation pictured in the fourth row consider a 3-disk D3 = D2×
[0, 1] that contains the portion of the branched locus that we want to change. Such a disk is depicted on the right-hand side of Figure1.6. The inverse image under the
n-fold covering of the 3-disk in the base manifold consists of a 3-disk D3
1 = D2×[0, 1]
as shown in the left-hand side of Figure1.6 and of n−3 other disjoint 3-disks, where
nis the order of the branched cover. On the disk D3
1, p restricts to ˜p×[0, 1] where ˜p is
the branched cover of Figure1.4while the other disks are mapped diffeomorphically into D3.
p
Figure 1.6:
If we cut S3 along the ∂D3 and we glue back the disk D3 with the identity
everywhere on its boundary except on D2× {1} where we use the diffeomorphism f
of Figure1.5. By Lemma 1.3.6 we still obtain a branched cover if we find covering diffeomorphisms on the boundary of the upstair disks. This is easy since for the diffeomorphically mapped disk we can use the same gluing map as downstair and for the disk D3
1 we use the identity everywhere except on D2× {1} where we use the
diffeomorphism g of Figure1.5.
1.4
Open Book Decompositions
In this section we introduce the notion of open book decomposition which will be of primary importance throughout the entire thesis because it interacts both with contact structures and with branched covers.
We begin with a variant of the definition of open book that is now often referred to as an abstract open book.
Definition 1.4.1. An abstract open book is a pair (F, φ) where
1. F is an orientable compact surface with non-empty boundary and
2. φ : F → F is a diffeomorphism such that φ is the identity in a neighborhood of ∂F . The map φ is called the monodromy.
Remark 1.4.2. Since φ is the identity in a neighborhood of ∂F , it is an orientation
preserving map.
Note that given an abstract open book (F, φ) we get a closed orientable 3-manifold M(F,φ) as follows.
Recall that the mapping torus F (φ) is the quotient space F ×[0, 2π]/ ∼ obtained by identifying (x, 2π) with (φ(x), 0) for each x ∈ F . Since φ is a diffeomorphism equal to the identity near the boundary ∂F , we have that F (φ) is in a natural way a differentiable manifold with boundary diffeomorphic to ∂F × S1.
We define M(F,φ) to be the manifold obtained by filling the (tori) boundary
components of F (φ) in the following way:
M(F,φ) := F (φ) ∪ψ
a |∂F |
S1× D2
,
where |∂F | denotes the number of boundary components of F and for each boundary component l of F , ψ is the unique (up to isotopy) diffeomorphism that takes S1×{p}
to [l×{0}], where p ∈ ∂D2 and takes {q}×∂D2 to [{q0} ×[0, 2π]] , where q ∈ S1 and
q0 ∈ ∂F. We denote the core of the solid tori in the definition of M(F,φ) by L(F,φ).
Definition 1.4.3. Two abstract open books (F1, φ1) and (F2, φ2) are called
equiv-alent if there is a diffeomophism h : F1 → F2 such that h−1 ◦ φ2 ◦ h and φ1 are
isotopic rel. boundary.
It is easy to see that equivalent abstract open books give diffeomorphic 3-manifolds. However we will be mainly interested in working with objects defined on a manifold only up to ambient isotopy. For that reason we want to define a similar object that already lives in a given manifold.
To do this it is important to note that there is a fibration
π(F,φ) : M(F,φ)\ L(F,φ) → S1 := R/2πZ, given by π(F,φ)([x, θ]) = [θ], for [x, θ] ∈ F (φ),
π(F,φ)(ϕ, reiθ) = [θ], for (ϕ, reiθ) ∈ ∂F ×
D2 \ {0}⊂ ∂F ×C \ {0}. This give rise to the following definition.
L
S1
Fθ
S1 π
Figure 1.7: Locally, around a point of the binding, every open book looks like this.
Definition 1.4.4. An open book decomposition of an closed orientable
3-manifold M consists of a link L, called the binding, and a fibration π : M \L → S1.
The fibres Fθ = π−1(θ), θ ∈ S1 := R/2πZ, are called the pages. Moreover, it is
required that each component of the binding L have a trivial tubular neighbourhood
L × D2 in which π is given by the angular coordinate in the D2–factor. The couple
(L, π) is called open book.
Remark1.4.5. The last condition on the tubular neighborhood is crucial. It basically
tells that the pages are Seifert surfaces of the binding. The interested reader may search on the web “rational open book” for examples of things that can happens without this hypothesis.
Example 1.4.6. Given an abstract open book (F, φ) we obtain an open book
de-composition (L(F,φ), π(F,φ)) on M(F,φ) as done just before Definition1.4.4.
Example 1.4.7. Regard S3 as the subset of C2 given by
S3 = {(z1, z2) ∈ C2 : |z1|2+ |z2|2 = 1}.
1. Set
U = {(z1, z2) ∈ S3 : z1 = 0},
and consider the fibration
πU : S3\ U −→ S1 ⊂ C
(z1, z2) 7−→
z1
|z1|
.
In polar coordinates this map is given by
This shows that (U, πU) is an open book with pages the 2–discs
πU−1(θ) = {|z2| ≤1, z1 = q
1 − |z2|2eiθ}.
This fibration is related to the well known fact that S3 is the union of two solid tori.
This open book in S3\ {pt} looks globally as Figure 1.7.
Figure 1.8: Different images of the same Hopf link. 2. Set
H = {(z1, z2) ∈ S3 : z1z2 = 0}.
This set H is the union of two Hopf fibres
Ki = {(z1, z2) ∈ S3 : zi = 0}, i = 1, 2.
The link H = K1t K2 is called the Hopf link, see Figure 1.8.
Consider the fibration
π+ : S3\ H −→ S1 ⊂ C
(z1, z2) 7−→
z1z2
|z1z2|
.
In polar coordinates this map is given by
π+: (r1eiθ1, r2eiθ2) 7−→ θ1+ θ2 ∈ S1 = R/2πZ.
The pages π−1
+ (θ) of the open book (H, π+) can be parametrized by
(0, 1) × S1 −→ π−1
+ (θ) ⊂ S 3\ H
(r0
, θ0) 7−→ (r0eiθ0,√1 − r02ei(θ−θ0)). (1.1)
This shows that (H, π+) is an open book decomposition with pages diffeomorphic
to annuli.
An annulus in S3 embedded as the one of this open book decomposition is called a
Hopf band. See Figure 1.9.
3. The Hopf link also admits another fibration, namely
π− : S3\ H −→ S1 ⊂ C
(z1, z2) 7−→
z1z2
|z1z2|
(H, π+) (H, π−)
Figure 1.9: A page of the open books (H, π+) and (H, π−).
In polar coordinates this map is given by
π−: (r1eiθ1, r2eiθ2) 7−→ θ1 − θ2 ∈ S1 = R/2πZ.
The pages π−1
− (θ) of the open book (H, π−) can be parametrized as
(0, 1) × S1 −→ π−1 + (θ) ⊂ S 3\ H (r0 , θ0) 7−→ (√1 − r02ei(θ+θ0) , r0eiθ0).
So we obtain another open book decomposition (H, π−) for S3.
For open book decomposition we have two notion of equivalence.
Definition 1.4.8. Two open books (L, π) and (L0, π0) on a 3-manifold M are
dif-feomorphicif there is a diffeomorphism H : M → M such that
(H(L), π ◦ H−1) = (L0
, π0).
If the diffeomorphism H preserves (or inverts) the orientation, the two open books are called positively (or negatively) diffeomorphic.
Similarly two open books (L, π) and (L0, π0) are isotopic if there is an isotopy
Ht: M × I → M starting from the identity such that
(H1(L), π ◦ H1−1) = (L 0
, π0).
Remark 1.4.9. The open books (L, π) and (L, −π) are diffeomorphic, but in general
they may be not isotopic.
One should note that it is important to include the projection in the data for an open book, since L does not determine the open book (even up to diffeomorphism or isotopy), as the following example shows.
Example 1.4.10. Let M = S1× S2 and L = S1× {N, S}, where N, S ∈ S2. There
are many ways to fiber M \ L = S1 × S1× I. In particular, if γ
n is an embedded
curve on the torus T = S1 × S1 in the homology class (1, n), then M \ L can be
fibered by annuli parallel to γn× I. There are diffeomorphisms of T that relate all
of these fibrations but the fibrations coming from γ0 and γ1 are not isotopic. There
On the other hand, any page F of an open book (L, π) completely determines
L = ∂F and also (though much less evidently, as seen in the work of Cerf [7],
Laudenbach and Blank [37] and Waldhausen [54]) ±π up to isotopy relative to F . Consider now an open book decomposition (L, π) in a closed orientable 3-manifold M. Choose an auxiliary Riemannian metric on M such that the vector field ∂θ on L × (D2 \ {0}) is orthogonal to the pages. Extend this to a vector field
on M \ L orthogonal to the pages that projects under the differential dπ to the vector field ∂θ on S1; this extends to a smooth vector field on M vanishing along
L. By slight abuse of notation, we write ∂θ for the vector field on M thus defined.
The time-2π-map of the flow of this vector field, that we call φ, sends every page to itself.
Definition 1.4.11. A diffeomorphism φ just defined is called a monodromy of the
open book decomposition (L, π).
Remark 1.4.12. Every two monodromy maps φ1 and φ2 of an open book
decom-position are equivalent, in the sense that φ1◦ φ−12 is isotopic rel. boundary to the
identity map.
Clearly abstract open books and (non-abstract) open book decompositions are closely related. The following lemma illustrate this relationship.
Lemma 1.4.13. We have the following basic facts about open books and abstract
open books:
1. An abstract open book (F, φ) determines a manifold M(F,φ) up to
diffeomor-phism and a well defined open book (L(F,φ), π(F,φ)) on it.
2. Equivalent abstract open books give diffeomorphic 3-manifolds.
3. An open book decomposition (L, π) of M determines an abstract open book
(F(L,π), φ(L,π)) such that M(F(L,π),φ(L,π)) is diffeomorphic to M with a
diffeomor-phism that sends (L(F(L,π),φ(L,π)), π(F(L,π),φ(L,π))) to (L, π).
Proof. 1. The fact that that an abstract open book (F, φ) determines M(F,φ) and
an open book (L(F,φ), π(F,φ)) was already noticed. The only choice we made in this
construction was that of the gluing map ψ of the boundary tori components, but two such maps are always isotopic so the resulting manifolds are diffeomorphic. 2. Let (F1, φ1) and (F2, φ2) be two abstract open books and h : F1 → F2 a
diffeo-morphism such that φ2 ◦ h = h ◦ φ1 (we suppose here that h−1◦ φ2 ◦ h and φ1 in
Definition 1.4.3 are equal and not only isotopic rel. boundary, however the general case is analogous). If we construct M(F1,φ1) using
ψ : a
|∂F1|
S1× D2 → ∂F1(φ1)
as gluing map on the solid tori we can suppose we build M(F2,φ2) by using the map
where ∂
(h × Id)/ ∼
denotes the restriction to the boundary of the quotient map (h × Id)/ ∼: F1(φ1) → F2(φ2). Therefore we can define a diffeomorphism
H: M(F1,φ1)→ M(F2,φ2) by the identity on the solid tori and
h ×Id : F1(φ1) → F2(φ2)
on the exterior of the solid tori.
3. Define F(L,π) to be a parametrizaion of the intersection of the page π−1(0)
(re-member, S1 = R/2πZ in the definition of open book), say, with the complement
of an open tubular neighbourhood L × Int(D2
1/2). A monodromy map of this open
book decomposition is the desired φ(L,π).
Now M(F(L,π),φ(L,π)) is easily seen to be diffeomorphic to M with a diffeomorphism
that sends L(F(L,π),φ(L,π)), π(F(L,π),φ(L,π))) to (L, π).
Remark 1.4.14. The abstract open book one obtains from an open book
decompo-sition as in part 3 of the lemma is by no means unique, however two different open books obtained in this way are equivalent.
Example 1.4.15. We calculate here the monodromy of the open books of Example
1.4.7.
1. For the first example, (U, πU) as a vector field ∂θ introduced just before Definition
1.4.11 we may take ∂θ1. This shows that (U, πU) is an open book with pages the
2–discs
πU−1(θ) = {|z2| ≤1, z1 = q
1 − |z2|2eiθ},
and monodromy the identity map.
2. Note that the collection of the parametrizations of equation1.1 for (H, π+), as θ
ranges over S1, gives a trivialization of the bundle π
+ : S3\ H → S1. Beware that
this does not imply that the monodromy map φ of this open book is the identity map.
Here is how to determine this monodromy map. Choose a smooth function [0, 1] → [0, 1], r 7→ δ(r) that is identically 1 near r = 0 and identically 0 near r = 1. Define a flow φt: S3 → S3 in polar coordinates (r1, θ1, r2, θ2) by
(
θ1 7→ θ1+ δ(r1)t
θ2 7→ θ2+ (1 − δ(r1))t.
Then π+◦ φt(z1, z2) = π+(z1, z2) + t, so this flow is transverse to the pages of
the open book and maps pages to pages. Moreover, near ri = 0 we have ˙φt = ∂θi,
i = 1, 2. This implies that the monodromy of the open book is given by the map φ2π. In terms of the parametrization of the pages by (r0, φ0) as in Example 1.4.7,
this map is, up to isotopy rel boundary, a right-handed Dehn twist along the core circle {r0 = 1/2} of the page, see Figure 1.10.
3.
The calculation for (H, π−) is similar. In terms of the parametrization of the pages
by (r0, φ0) as in Example1.4.7, the monodromy map is, up to isotopy rel boundary,
∂θ
r1 = 0 r1 = 1
Figure 1.10: A right-handed Dehn twist.
Remark 1.4.16. Note that when we discuss open books we consider the binding and
pages up to isotopy in M, whereas when we discuss abstract open books we can only consider them up to diffeomorphisms. In Chapter 3 this distinction is crucial an we restrict ourselves to (non-abstract) open books.
We have thus obtained a closed orientable 3-manifold and an open book decom-position on it for every abstract open book. The following old theorem by Alexander, originally stated in terms of abstract open books, shows the converse.
Theorem 1.4.17 (Alexander 1923, [1]). Every closed orientable 3-manifold M has
an open book decomposition (L, π).
Proof. By Theorem1.2.10every closed orientable 3-manifold M is a branched cover
of S3 with branch set a link L0. By a result also contained in [1] every link can be
braided about the unknot U, that if S1 × D2 = S3 \ U then we can isotope L0 so
that L0 ⊂ S1 × D2 and L0 is transverse to {p} × D2 for all p ∈ S1.
Let p : M → S3 be the branched covering map. Set L = p−1(U) ⊂ M. Then, L
is the binding of an open book. In fact, the fibering of the complement of B is simply
π = πU ◦ p, where πU is the fibering of the complement of U in S3. The condition
on the tubular neighborhood is satisfied since L does not belong to p−1(L0).
1.5
Covers of Open Book Decompositions
Theorem 1.4.17 shows that every branched cover gives rise to an open book on the upstairs manifold. As a sort of converse we have this beautiful theorem by Morton and Montesinos. The reader should be aware of the fact that the relevance of this theorem is due to the optimal bounds on the number of sheets.
Theorem 1.5.1 (Montesinos and Morton 1991, [43]). Let (L, π) be any open-book
decomposition of a closed 3-manifold manifold M. Then there are a closed braid C ⊂ S3 with axis U , and a d-sheeted simple branched cover p: M → S3 with branch
set C such that Fθ = p−1(Dθ) for each θ ∈ S1, where {Dθ} are the disk pages of the
standard open book (U, πU) of S3.
The binding L of the decomposition is then p−1(U). If L has k components, we can choose p so that d= max{k, 3}.
Sketch of the proof. Let F be a surface diffeomorphic to a page of (L, π). The
au-thors show that there exist a d-fold simple branched cover r : F → D2, where
d = max{k, 3}. They also show that one can realize the monodromy of the open
book decomposition as the lift of a braid (viewed as a diffeomorphism of the disk
D2 fixing ∂D2 and the branch set in D2) under the branched cover r. Now consider
the closure of this braid C (viewed as a geometric object in the usual sense) in S3
with respect to an axis U. Endow S3 with the open book decomposition with disks
Dθ as pages, U = ∂Dt as binding, and the identity map as monodromy. This
con-struction gives a d-fold branched cover p : M → S3 such that each page of the given
open book decomposition is realized as p−1(D
θ) and the binding is simply equal to
Contact Topology
In this chapter we recall many basic and some advanced facts about contact topology that will be used in the next chapters. It is not meant to be an introductory chapter for someone who is willing to properly study this subject since we won’t prove anything and many important basic aspects are omitted. However we tried to present the material in a clear way, that hopefully will be accessible to an undergraduate student with basic knowledge on smooth manifolds.
For a complete introductory reference on the subject, from which we have taken most of the statements of this chapter, see the book "An Introduction to Contact Topology" by Geiges [24].
2.1
Contact Manifolds
In this section we introduce basic definition and examples. After that we show that many important theorems of differential topology, such as the disk theorem or the isotopy extension theorem, have a "contact" version.
2.1.1
Examples of Contact Manifolds
Let M be a differential 3-manifold, T M its tangent bundle, and ξ ⊂ T M a plane field on M, that is, a smooth sub-bundle of codimension 1. In order to describe special types of plane fields, it is useful to present them as the kernel of a differential 1–form.
Lemma 2.1.1. Locally, ξ can be written as the kernel of a differential 1–form α.
It is possible to write ξ = Ker α with a 1–form α defined globally on all of M if and only if ξ is coorientable, which by definition means that the quotient line bundle T M/ξ is trivial.
In the following we shall always assume our plane fields ξ to be coorientable. One class of plane fields that has received a great deal of attention are the integrable ones. This term denotes plane fields with the property that through any point p ∈ M one can find a surface N whose tangent spaces coincide with the plane field, i.e. such that TqN = ξq for all q ∈ N. Such an N is called an integral surface
of ξ. It turns out that ξ = Ker α is integrable precisely if α satisfies the Frobenius integrability condition
In terms of Lie brackets of vector fields, this integrability condition can be written as
[X, Y ] ∈ ξ for all X,Y ∈ ξ;
here X ∈ ξ means that X is a smooth section of T M with Xp ∈ ξp for all p ∈ M. A
third equivalent formulation of integrability is that ξ is locally of the form dz = 0, where z is a local coordinate function on M. The collection of integral surfaces of an integrable plane field constitutes what is called a 2-dimensional foliation.
Contact structures are in a certain sense the exact opposite of integrable plane fields.
Definition 2.1.2. Let M be a 3-manifold. A contact structure is a maximally
non-integrable plane field ξ = Ker α ⊂ T M, that is, the defining differential 1–form
α is required to satisfy
α ∧ dα 6= 0
(meaning that it vanishes nowhere). Such a 1–form α is called a contact form. The pair (M, ξ)is called a contact manifold.
Remark2.1.3. As a somewhat degenerate case, this definition includes 1-dimensional
manifolds with a non-vanishing 1–form α. The corresponding contact structure
ξ= Ker α is the zero section of the tangent bundle.
Example 2.1.4. On R3 with Cartesian coordinates (x, y, z), the 1–form
αstd = dz + xdy
is a contact form. The contact structure ξstd = Ker αstd is called the standard
contact structureon R3. See Figure 2.1.
x
y z
Remark 2.1.5. Observe that α is a contact form precisely if α ∧ dα is a volume form
on M (i.e. a nowhere vanishing 3-dimensional differential form); in particular, M needs to be orientable. The condition α∧dα 6= 0 is independent of the specific choice of α and thus is indeed a property of ξ = Ker α: any other 1–form defining the same hyperplane field must be of the form λα for some smooth function λ : M → R \ {0}, and we have
(λα) ∧ d(λα) = λα ∧ (λdα + dλ ∧ α) = λ2
α ∧ dα 6= 0.
Definition 2.1.6. A contact structure ξ on M is cooriented if it comes with a
prescribed orientation of the quotient bundle T M/ξ. Similarly ξ is oriented if there is an orientation on ξ.
Remark 2.1.7. When we speak of orientation of vector bundles we mean that the
fibers are oriented, not the bundles as a manifolds.
Since M is oriented, there is a canonical way to orient a cooriented contact structure and viceversa: just orient (or coorient) it in such a way that a positive base for ξ (or T M/ξ) followed by a positive base for T M/ξ (or ξ) gives the orientation of M (there is a little bit of abuse of notation in this sentence but hopefully we gained clarity). Therefore, using this convention, we will use the terms oriented or cooriented contact structure as synonymous.
Lemma/Definition 2.1.8. Associated with a contact form α one has the so-called
Reeb vector field Rα , uniquely defined by the equations
• dα(Rα, −) ≡ 0,
• α(Rα) ≡ 1.
Notice that one cannot reasonably speak of the Reeb vector field of a contact structure.
The name ‘contact structure’ has its origins in the fact that one of the first his-torical sources of contact manifolds are the so-called spaces of contact elements.
Lemma/Definition 2.1.9. A Liouvil le vector field Y on a symplectic
4-manifold (W, ω) is a vector field satisfying the equation LYω = ω, where L denotes
the Lie derivative. In this case, the 1–form α := iYω := ω(Y, −) is a contact form
on any3-manifold M transverse to Y (that is, with Y nowhere tangent to M). Such
3-manifolds are said to be of contact type.
Example 2.1.10. Given a contact manifold (M, ξ = Ker α), one can form the
symplectic manifold (R × M, ω = d(etα)), with t denoting the R–coordinate. (Here
α is interpreted as a 1–form on R × M, i.e. we identify α with its pull-back under
the projection R × M → M). Indeed,
ω2 = (et(dt ∧ α + dα))2 = 2e2tdt ∧ α ∧ dα 6= 0.
One easily sees that the vector field ∂t is a Liouville vector field for this symplectic
form ω. The manifold (R × M, ω) is called the symplectisation of (M, α). The orientation of R × M induced by the volume form ω2 coincides with the product
Example 2.1.11. Let (x1, y1, x2, y2) be Cartesian coordinates on R4. Then the
standard contact structure on the unit sphere S3 in R4 is given by the contact
form
αstd := xdy − ydx.
Write r for the radial coordinate on R4, that is, r2 = x2
1+ y12+ x22+ y22. One checks
easily that rdr ∧ αstd∧ dαstd 6= 0 for r 6= 0. Since S3 is a level surface of r (or r2),
this verifies the contact condition.
Alternatively, consider W = R4 with its standard symplectic form
ω= dx1∧ dy1+ dx2∧ dy2.
The radial vector field
Y = 1
2r∂r = 1
2(x1∂x1 + y1∂y1 + x2∂x2 + y2∂y2)
(where r2 = x2
1 + y12+ x22+ y22) is a Liouville vector field for ω. On the unit sphere
S3 ⊂ R4 it induces, by Lemma 2.1.9, the contact form
1
2αstd= iYω=
1
2(x1dy1− y1dx1+ x2dy2− y2dx2).
This again shows that αstd is a contact form on that sphere.
Here is a further example of a contact form on R3.
Example 2.1.12. On R3, with (r, ϕ) denoting polar coordinates in the (x, y)–plane,
the following 1–form is a contact form:
αsym := dz + r2dϕ= dz + xdy − ydx.
This contact structure is called the symmetric contact structure, see Figure2.2. In fact, the contact form of the previous example is not really ‘different’ from the standard contact form αstd. The following definition gives a precise notion for
the equivalence of contact structures or forms.
Definition 2.1.13. Two contact manifolds (M1, ξ1) and (M2, ξ2) are said to be
contactomorphic if there is a diffeomorphism f : M1 → M2 with df(ξ1) = ξ2. If
ξi = Ker αi, i = 1, 2, this is equivalent to saying that α1 and f∗α2 determine the
same plane field, and hence equivalent to the existence of a nowhere zero function
λ : M1 → R \ {0} such that f∗α2 = λα1. Occasionally one speaks of a strict
contactomorphismbetween the strict contact manifolds (M1, α1) and (M2, α2)
if f∗α
2 = α1.
Example 2.1.14. The contact manifolds (R3, ξstd = Ker αstd) and (R3, ξsym =
Ker αsym), from Example 2.1.4 and 2.1.12 are contactomorphic. An explicit
con-tactomorphism f with f∗α
sym = αstd is given by
f(x, y, z) =(x + y)/2, (y − x)/2, z + xy/2.
x
y z
Figure 2.2: The symmetric contact structure.
Example 2.1.15. On R3 with cylindrical coordinates (r, ϕ; z), define the 1–form
αot = cos r dz + r sin r dϕ.
It is easy to see that αot is a contact form (although for the computation is better
to do a coordinate change first). The contact structure ξot = Ker αot is called the
standard overtwisted contact structureon R3.
Observe that both the standard contact structure ξsym and the standard
over-twisted contact structure ξot are horizontal along the z-axis {r = 0}, and all the
rays perpendicular to the z–axis (with z and ϕ constant) are tangent to both ξstd
and ξot . For r 6= 0, the contact structure ξstd is spanned by the vector fields ∂r
and ∂ϕ− r2∂z; the contact structure ξot is spanned by ∂r and cos r ∂ϕ− rsin r ∂z.
So both ξstd and ξot turn counterclockwise as one moves outwards from the z–axis
along any such ray. However, while the rotation angle of ξstd approaches (but never
reaches) π/2 (i.e. the contact planes never become vertical), the contact planes of
ξot make infinitely many complete turns as one moves out in radial direction, see
Figure2.3.
Bennequin [3] proved that (R3, ξ
std) and (R3, ξot) are in fact not contactomorphic.
This result can be regarded as the birth of contact topology, since it tells us that the classification of contact structures is not purely topological, that is, it is not only a matter of obstruction theory. See [48] for a beautiful survey on the classification of non-integrable distributions.
Here is a further useful example of contactomorphic manifolds.
Proposition 2.1.16. For any point p ∈ S3, the two contact manifolds(S3\{p}, ξ
std)
and (R3, ξ
x
y z
Figure 2.3: The standard overtwisted contact structure.
Remark 2.1.17. Since also (R3, ξ
sym) and (R3, ξstd) are contactomorphic, this clarifies
why we have used the same symbol ξstd for the standard contact structures on R3
and S3.
Example 2.1.18. There is a rather less contrived contactomorphism between
(R3, ξ
sym) and the lower hemisphere in (S3, ξstd) given by
(x, y, z) 7→ √ (x, y, z, −1) 1 + x2+ y2+ z2.
2.1.2
Gray Stability Theorem
The Gray stability theorem that we are going to state now says that there are no non-trivial deformations of contact structures on closed manifolds. In fancy language, this means that contact structures on closed manifolds have discrete moduli.
Theorem 2.1.19 (Gray stability theorem). Let ξt, t ∈[0, 1], be a smooth family of
contact structures on a closed 3-manifold M. Then there is an isotopy (Ψt)t∈[0,1] of
M such that
dΨt(ξ0) = ξt for each t ∈ [0, 1].
Beware that contact forms do not satisfy stability, that is, with αt a smooth
family of contact forms one cannot, in general, find an isotopy Ψt such that Ψ∗tαt =
α0.
Remark 2.1.20. The condition in Theorem 2.1.19that M be a closed manifold is
es-sential. Eliashberg [12] has shown that there are non-trivial deformations of contact structures on S1
× R2.
2.1.3
Darboux’s Theorem and Neighborhoods Theorems
The flexibility of contact structures, which finds its expression in the Gray stability theorem, results in a variety of theorems that can be summed up as saying that
there are no local invariants in contact geometry. We present now some of those theorems that will be useful for the scopes of this thesis.
In contrast with Riemannian geometry, for instance, where the local structure coming from the curvature gives rise to a rich theory, the interesting questions in contact geometry thus appear only at the global level. However, it is actually the local flexibility that allows us to prove strong global theorems, such as the existence of contact structures on every closed orientable 3-manifolds.
Darboux’s Theorem
Theorem 2.1.21 (Darboux’s theorem). Let α be a contact form on the 3–manifold
M and p a point on M . Then there are coordinates x, y, z on a neighbourhood U ⊂ M of p such that p= (0, 0, 0) and
α|U = dz + xdy.
Remark 2.1.22. Observe that the map (x, y, z) 7→ (x, y, 2z) is a contactomorphism
of the standard contact structure ξstd on R3 for any ∈ R+. Therefore it is an
immediate consequence of the Darboux’s Theorem that there is a contact embedding of the closed unit ball Bstin (R3, ξstd) into (M, ξ = Ker α) which sends the origin to p.
Here ‘contact embedding of Bst’ simply means a contactomorphism of a small open
neighbourhood of Bst in (R3, ξstd) onto its image in (M, ξ); later we shall encounter
a more general concept of contact embedding.
In fact, by Proposition 2.1.16 and Example 2.1.18 there is a contactomorphism of (R3, ξ
std) with a relatively compact subset of itself, and hence by scaling with a
subset of Bst. So we can also construct a contactomorphism between (R3, ξstd) and
a neighbourhood of p in (M, ξ).
Theorem 2.1.23. Given a compact subset C of an oriented3-manifold M, and two
contact structures ξ0 and ξ1 on M that agree on C, there exist neighborhoods U0 and
U1 of C, and an isotopy ψt of M that fixes C, so that ψ1 : (U0, ξ0) → (U1, ξ1) is a
contactomorphism.
Isotropic Submanifolds
Definition 2.1.24. Let (M, ξ) be a contact 3-manifold. A submanifold L of (M, ξ)
is called an isotropic submanifold if TpL ⊂ ξp for all points p ∈ L. A 1-dimension
isotropic submanifold is called a Legendrian submanifold
Theorem 2.1.25. Let (Mi, ξi), i = 0, 1, be contact 3-manifolds with closed
Legen-drian submanifolds (L0i, ξi0). Suppose there is a diffeomorphism φ : L01 → L0
2. Then
φ extends to a contactomorphism of suitable neighborhoods N(L0i) of L0i.
Corollary 2.1.26. Every Legendrian submanifold K diffeomorphic to S1 in a
con-tact 3-manifold (M, ξ) has a neighborhood contactomorphic to
S1× R2, Ker(cos θ dx + sin θ dy),
Contact Submanifolds
Definition 2.1.27. Let (M, ξ) be a contact 3-manifold. A submanifold M0 of M
with contact structure ξ0is called a contact submanifold of (M, ξ) if T M0∩ξ|
M0 =
ξ0.
Remark 2.1.28. A 1–dimensional manifold M0 with its unique contact structure (see
Remark 2.1.3) is a contact submanifold of (M, ξ) if and only if M0 is transverse to
ξ. A case of particular interest to us will be transverse links in contact 3–manifolds.
Theorem 2.1.29. Let(Mi, ξi), i = 0, 1, be contact 3-manifolds with compact contact
submanifolds(Mi0, ξi0). Suppose there is a contactomorphism φ : (M10, ξ10) → (M20, ξ20). Then φ extends to a contactomorphism of suitable neighborhoods N(Mi0) of Mi0.
Corollary 2.1.30. Every contact submanifold K diffeomorphic to S1 in a contact
3-manifold (M, ξ) has a neighborhood contactomorphic to
S1× R2, Ker(dθ + xdy − ydx),
where θ denotes the S1 coordinate and K is identified with S1× {0}.
Surfaces
A classical method for studying 3-dimensional contact structures relies on their interaction with embedded surfaces. Let S be a surface inside the oriented contact manifold (M, ξ). Then ξ|S and T S are two plane fields on S. We call a point p
singular if ξp = TpS. Note that because of the non-integrability of ξ, the singular set
Qis of dimension one or less. In fact, for generic surfaces it will consist of a discrete
collection of points. Away from the singular points, the intersection of ξ and T S define a line field on S. Locally, we may find integral curves for this line field, and every point in the surface is either singular, or contained in one of these curves. We use this to define a singular foliation on S.
Definition 2.1.31. The characteristic foliation Sξ of a surface S in (M, ξ) is the
singular 1–dimensional foliation of S defined by the distribution (T S ∩ ξ|S).
Figure 2.4: The characteristic foliation on S2 ⊂(R2, ξ sym).
Locally we may find sections of this line bundle, i.e. vector fields tangent to the leaves of Sξ. If S and ξ are oriented, then we may do better.
