# The homotopy type of the contactomorphism groups of tight contact
$3$-manifolds, part I

Eduardo Fernández  Universidad Complutense de Madrid, Facultad de Matemáticas,
and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, C. Nicolás Cabrera,
13-15, 28049 Madrid, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>,
Javier Martínez-Aguinaga  Universidad Complutense de Madrid, Facultad de
Matemáticas, and Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, C.
Nicolás Cabrera, 13-15, 28049 Madrid, Spain.
<EMAIL_ADDRESS>and  Francisco Presas  Instituto de Ciencias
Matemáticas CSIC-UAM-UC3M-UCM, C. Nicolás Cabrera, 13-15, 28049 Madrid, Spain.
<EMAIL_ADDRESS>
###### Abstract.

We prove that that the homotopy type of the path connected component of the
identity in the contactomorphism group is characterized by the homotopy type
of the diffeomorphism group plus some data provided by the topology of the
formal contactomorphism space. As a consequence, we show that every connected
component of the space of Legendrian long knots in ${\mathbb{R}}^{3}$ has the
homotopy type of the corresponding smooth long knot space. This implies that
any connected component of the space of Legendrian embeddings in
${\mathbb{S}}^{3}$ is homotopy equivalent to the space
$K(G,1)\times\operatorname{U}(2)$, with $G$ computed by A. Hatcher and R.
Budney. Similar statements are proven for Legendrian embeddings in
${\mathbb{R}}^{3}$ and for transverse embeddings in ${\mathbb{S}}^{3}$.
Finally, we compute the homotopy type of the contactomorphisms of several
tight $3$-folds: ${\mathbb{S}}^{1}\times{\mathbb{S}}^{2}$, Legendrian
fibrations over compact orientable surfaces and finite quotients of the
standard $3$-sphere. In fact, the computations show that the method works
whenever we have knowledge of the topology of the diffeomorphism group. We
prove several statements on the way that have interest by themselves: the
computation of the homotopy groups of the space of non-parametrized
Legendrians, a multiparametric convex surface theory, a characterization of
formal Legendrian simplicity in terms of the space of tight contact structures
on the complement of a Legendrian, the existence of common trivializations for
multi-parametric families of tight ${\mathbb{R}}^{3}$, etc.

###### 2020 Mathematics Subject Classification:

Primary: 53D35, 57K33.

###### Contents

  1. 1 Introduction.
    1. 1.1 Outline.
    2. 1.2 Acknowledgements.
  2. 2 Preliminaries.
    1. 2.1 Convex surface theory in tight contact $3$–manifolds.
    2. 2.2 Fibrations in contact topology.
    3. 2.3 The space of Darboux balls in a contact manifold.
    4. 2.4 Legendrian embeddings in $({\mathbb{S}}^{3},\xi_{\operatorname{std}})$.
    5. 2.5 Smooth Case. Hatcher’s Theorems.
    6. 2.6 Microfibration Lemma
  3. 3 Eliashberg and Giroux Theorems
    1. 3.1 Classification of tight contact structures in the $3$–sphere.
    2. 3.2 Contactomorphisms in the standard contact $3$–sphere.
    3. 3.3 Standard convex disks on a tight contact $3$-fold.
    4. 3.4 The space of standard convex spheres in a tight $3$-fold.
  4. 4 Reduction relative to a Giroux-handlebody.
    1. 4.1 Contact cell decompositions.
    2. 4.2 Main reduction Theorem.
    3. 4.3 Deformation at the $0$–skeleton.
    4. 4.4 Deformation at the $1$–skeleton.
    5. 4.5 The failure of the non-relative case.
  5. 5 Proofs.
    1. 5.1 Proof of Theorem 1.0.4.
    2. 5.2 Proof of Theorem 1.0.1.
    3. 5.3 Proof of Theorem 1.0.2.
    4. 5.4 Proof of Theorem 1.0.3.
  6. 6 Applications.
    1. 6.1 Contactomorphism groups of some tight contact $3$–folds.
    2. 6.2 Transverse embeddings.
    3. 6.3 Formal Legendrians and the generalized Thurston–Bennequin inequality.
    4. 6.4 The homotopy type of the space of tight contact structures of Legendrian complements.
    5. 6.5 The space of Legendrian unknots.
    6. 6.6 The space of positive Legendrian torus knots.
    7. 6.7 The space of Legendrian parametrized $(n,n)$-torus links.
    8. 6.8 Legendrian embeddings in $({\mathbb{R}}^{3},\xi_{\operatorname{std}})$.
    9. 6.9 The space of embedded Legendrians.
    10. 6.10 Tight contact structures on ${\mathbb{R}}^{3}$.

##  1\. Introduction.

Notations

  * •

${\mathbb{B}}^{n}(r)\subseteq{\mathbb{R}}^{n}$ will denote the $n$-dimensional
open ball of radius $r$ and
${\mathbb{D}}^{n}(r)=\overline{{\mathbb{B}}}^{n}(r)$ will stand for the
$n$-dimensional disk. The superindex $n$ will be omitted when it is clear that
we are referring to a $n$-dimensional ball/disk. Moreover, we will write
${\mathbb{B}}^{n}$ and ${\mathbb{D}}^{n}$ to denote the radius $1$ ball/disk.

  * •

We will usually denote by $z\in{\mathbb{S}}^{k}$, $k\geq 0$, the coordinate in
the sphere ${\mathbb{S}}^{k}$; and by
$(z,r)\in{\mathbb{D}}^{k+1}={\mathbb{S}}^{k}\times[0,1]/\sim$,
$z\in{\mathbb{S}}^{k}$ and $r\in[0,1]$ the polar coordinates in the
${\mathbb{D}}^{k+1}$ disk. Recall that $(z,0)$ is the centre of the disk for
any $z\in{\mathbb{S}}^{k}=\partial{\mathbb{D}}^{k+1}$ and
$z=(z,1)\in{\mathbb{S}}^{k}$.

  * •

${\mathcal{O}p}(A)$ will be denoting an arbitrarily small neighborhood of the
set $A$.

  * •

$\operatorname{Diff}(M)$ will refer to the group of diffeomorphisms of a
manifold $M$. The manifolds will be either closed or with boundary. In the
second case, we will be referring to the group of diffeomorphisms that are the
identity on ${\mathcal{O}p}(\partial M)$. In the rare event that we are
speaking of the open manifold, we will write it clearly.

  * •

$\operatorname{Cont}(M,\xi)$ will refer to the group of contactomorphisms of a
manifold $M$ with contact structure $\xi$. The same previous discussion for
the case of boundaries and open manifolds applies here.

  * •

The same works for $\operatorname{FCont}(M,\xi)$ that stands for the group of
formal contactomorphisms.

  * •

Informal notation: when we say that a diffemorphism has compact support, we
really mean that is the identity on ${\mathcal{O}p}(\partial M)$. We
understand that this name in the literature is reserved for the case of
functions, but we believe that it is a reasonable naming when speaking of
diffeomorphisms (or formal contactomorphisms or contactomorphisms).

  * •

For any of the previous groups, let us fix the contactomorphisms case as an
example, $\operatorname{Cont}(M,\xi;\operatorname{rel}C)$ means the space of
contactomorphisms on the closed manifold $M\setminus{\mathcal{O}p}(C)$, for
$C\subseteq M$. It should be understood as usual as manifold with boundary.

  * •

Though, most of our results, should apply to the non co-orientable case, when
we speak of a contact structure, we are assuming that is a co-oriented one and
that the contactomorphisms preserve the co-orientation.

  * •

We will denote by ${\mathfrak{C}Str}(M)$ the space of co-oriented contact
structures on $M$. If $(M,\xi)$ is a contact manifold
${\mathfrak{C}Str}(M,\xi)$ is the path–connected of $\xi$ inside
${\mathfrak{C}Str}(M)$.

  * •

${\mathfrak{Emb}}(N,M)$ is the space of proper embeddings of a manifold $N$
into a fixed manifold $M$, most of the times $N={\mathbb{S}}^{1}$ and
$M={\mathbb{S}}^{3}$ or $M={\mathbb{R}}^{3}$. When we refer to a particular
connected component based on a specific embedding $e$, we will write
${\mathfrak{Emb}}^{e}(N,M)$. If $\partial N\neq\emptyset$ isotopies in
${\mathfrak{Emb}}^{e}(N,M)$ are always assumed to be fixed at
${\mathcal{O}p}(\partial N)$. In the case $N={\mathbb{S}}^{1}$ if we restrict
to the subspace of embeddings that fix the image at $t=0$ to be $p\in M$, and
$\gamma^{\prime}(0)=v$, $v\in T_{p}M$, we will denote this subspace by
${\mathfrak{Emb}}_{p,v}({\mathbb{S}}^{1},M)$. The most frequent appearance of
this notation is in
${\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$, this is the
space of knots that pass through the North pole $N=(1,0)\in{\mathbb{C}}^{2}$
and recalling that $T_{N}{\mathbb{S}}^{3}=\\{iN,jN,kN\\}$, we fix the
direction $jN=(0,1)$. Here $i,j,k$ stands for the usual notation for the
quaternions in ${\mathbb{R}}^{4}={\mathbb{C}}^{2}$. The choice of tangent
vector is not casual, as we will see that notation is also used for Legendrian
knots and in that case (with standard contact structure $jN$ is a Legendrian
vector, in fact $iN$ is the Reeb.

  * •

The same notations apply to the space of Legendrian embeddings
${\mathfrak{Leg}}({\mathbb{S}}^{1},M)$. We mean that we fix position and
vector by using the analogous convention, we fix a connected component in the
same way, etc.

  * •

It is important to note that otherwise is stated our knots will be
parametrized. In Subsection 6.9 we will explain some facts about the non-
parametrized case.

Let us not speak of contactmorphisms for a while. We would like to discuss
where everything started. The study of the homotopy type of the space of
Legendrian embeddings of a contact manifold has become a central problem in
Contact Topology. The most studied case was the classification of the
connected components, the Legendrian knots. The tactic was always the same: to
try to relate it with the connected components of the space of formal
Legendrian knots. For instance, formal Legendrian knots are classified in
${\mathbb{S}}^{3}$ by three invariants: the smooth knot type of the embedding
and two integer valued invariants: the rotation invariant $\operatorname{rot}$
and the Thurston-Bennequin invariant $\operatorname{tb}$. So a natural
question is whether the natural map that assigns to each Legendrian embedding
$\gamma$ the three objects: smooth knot type, $\operatorname{rot}$ and
$\operatorname{tb}$ is a bijection. The first result in that line was the
Thruston-Bennequin inequality [Ben83]. It was proven that for the standard
contact structure in ${\mathbb{S}}^{3}$ (later on for any tight $3$-fold) the
previous map was not surjective. The reason is that the $3$ invariants satisfy
an inequation. In a sense, one of the ways of understanding the next
discussion is to say that somehow we prove a Thurston-Bennequin equation for
higher rank homotopy groups. To show whether the map was injective was next in
line. The first examples of non-injectivity were for Legendrian knots in
${\mathbb{S}}^{1}\times{\mathbb{S}}^{2}\\#{\mathbb{S}}^{1}\times{\mathbb{S}}^{2}$
[Fra96, DG10]. They were very much based in Legendrian surgery plus a
combination of the clasification of tight contact structures in some
$3$-folds. The biggest step forward came with the introduction of Legendrian
homology [Che02, EGH00], a Floer type invariant based on holomorphic curves. A
whole industry appeared showing either the non injectivity or the non
surjectivity of the map [Ng03, CN13, Etn05a]. It was generalized in several
directions: examples for more general $3$–folds [Sab03], higher dimensional
manifolds [EES07], etc.

Strangely enough, when the search for examples in higher homotopy groups
started, it was shown that all the known examples in ${\mathbb{R}}^{3}$, using
the natural generalization of the Legendrian homology, were, in fact, non
contractible in the space of formal Legendrians. Denote

$i_{F}:{\mathfrak{Leg}}({\mathbb{R}}^{3},\xi_{\operatorname{std}})\to{\mathfrak{FLeg}}({\mathbb{R}}^{3},\xi_{\operatorname{std}}),$

the paper [K0́5] was able to find examples of homotopy classes
$\\{\gamma_{\theta}\\}_{\theta\in{\mathbb{S}}^{1}}$ of the space
${\mathfrak{Leg}}({\mathbb{R}}^{3},\xi_{\operatorname{std}})$ that were
trivial as elements in the space of differentiable embeddings
${\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{R}}^{3})$. However, it was shown
that its image
$(i_{F})_{*}(\\{\gamma_{\theta}\\}_{\theta\in{\mathbb{S}}^{1}})$ in the
corresponding homotopy group of the formal Legendrian space
$\pi_{1}({\mathfrak{FLeg}}({\mathbb{R}}^{3},\xi_{\operatorname{std}}))$ was
also not trivial [FMP17]. This article shows that the opposite is true: the
map $i_{F}$ that embeds Legendrian knots in formal Legendrian knots induces
morphisms $(\pi_{k})_{*}(i_{F})$ that are injective for $k>0$ (see Theorem
6.3.2). But, we show much more: most of the formal Legendrian knot classes do
not admit a Legendrian representative.

However, we get to these results by a trick due to Hatcher, and further
refined by Budney, to study the homotopy type of the space of knots in the
$3$-sphere [Bud10, Hat99b, Hat]. They compare these spaces with the spaces of
diffeomorphisms of the complementary of each knot. The key observation is that
in order to understand the homotopy type of the space of smooth embeddings it
is enough to understand the space of long embeddings

${\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})=\\{\gamma\in{\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{S}}^{3}):\gamma(0)=N,\gamma^{\prime}(0)=jN\\}.$

Indeed, we can construct a fibration

(1)
${\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})\hookrightarrow{\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{S}}^{3})\rightarrow
V_{4,2}=\operatorname{SO}(4)/\operatorname{SO}(2).$

Moreover, the group of diffeomorphisms $\operatorname{Diff}({\mathbb{D}}^{3})$
of the $3$–disk relative to an open neighbourhood of the boundary naturally
acts over the path–connected component of the space of long embeddings
${\mathfrak{Emb}}_{N,jN}^{\gamma}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$
containing a fixed long embedding $\gamma$. The stabilizer of a long embedding
$\gamma$ is the group of diffeomorphisms $\operatorname{Diff}(C_{\gamma})$ of
the knot complement
$C_{\gamma}={\mathbb{S}}^{3}\backslash{\mathcal{O}p}(\gamma)$ that fix an open
neighbourhood of the boundary. Thus, there is a locally trivial fiber bundle

(2)
$\operatorname{Diff}(C_{\gamma})\hookrightarrow\operatorname{Diff}({\mathbb{D}}^{3})\rightarrow{\mathfrak{Emb}}_{N,jN}^{\gamma}({\mathbb{S}}^{1},{\mathbb{S}}^{3}).$

In the contact setting the situation is pretty similar. The homotopy type of
the space of Legendrian embeddings in the standard
$({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ is determined by the
corresponding space of long Legendrian embeddings
${\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})={\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})\cap{\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}}).$
In fact, the restriction of the fibration map (1) to the space of Legendrian
embeddings provides a fibration

(3)
${\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},{\operatorname{std}})\hookrightarrow{\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\rightarrow\operatorname{U}(2).$

As in the smooth case we also have a locally trivial fiber bundle

(4)
$\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}})\hookrightarrow\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})\rightarrow{\mathfrak{Leg}}_{N,jN}^{\gamma}({\mathbb{S}}^{3},\xi_{\operatorname{std}}),$

where
${\mathfrak{Leg}}_{N,jN}^{\gamma}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
denotes the path–connected component of the space of long Legendrian
embeddings containing a fixed Legendrian embedding $\gamma$.

Consider the natural inclusions of the contact fibration (4) inside the smooth
one (2), this produces the following commutative diagram

$\textstyle{\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Leg}}_{N,jN}^{\gamma}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diff}(C_{\gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diff}({\mathbb{D}}^{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Emb}}_{N,jN}^{\gamma}({\mathbb{S}}^{1},{\mathbb{S}}^{3}).}$

Realize that the groups $\operatorname{Diff}({\mathbb{D}}^{3})$ and
$\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})$ are both
contractible. This is due to Hatcher [Hat83] and Eliashberg [Eli92]. One
comment is in order, for several years Eliashberg’s proof was thought to be
correct, but with several technical holes on the higher rank computation. This
has been sorted out by D. Jänichen in [Jän18]. In fact, we believe that with
the current state of the art several proofs can be probably arranged: the one
already mentioned just works over Eliashberg’s original argument, with our
techniques we may provide a proof by reducing the problem to the
diffeomorphism group of ${\mathbb{S}}^{3}$, i.e. Hatcher’s theorem, finally we
also believe that the natural reformulation of the convex surface theory in
the high dimensional case [HH18, HH19] may provide a simpler proof of this
classical result. Anyhow, for the time being, we will assume this result.
Thus, for any $k\geq 0$ we have the following commutative square

$\textstyle{\pi_{k+1}({\mathfrak{Leg}}_{N,jN}^{\gamma}({\mathbb{S}}^{3},\xi_{\operatorname{std}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{k}(\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{k+1}({\mathfrak{Emb}}_{N,jN}^{\gamma}({\mathbb{S}}^{1},{\mathbb{S}}^{3}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{k}(\operatorname{Diff}(C_{\gamma})),}$

where the horizontal arrows are group isomorphisms.

We will prove

######  Theorem 1.0.1.

Let $\gamma\in{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ be a
long Legendrian embedding. The natural inclusion
$\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}})\hookrightarrow\operatorname{Diff}(C_{\gamma})$
induces a homotopy equivalence.

A relevant consequence of Theorem 1.0.1 is a relative version of Eliashberg
classification of tight contact structures on the $3$–ball [Eli92]. See
Theorem 6.4.6. Let us extract more corollaries of it.

######  Theorem 1.0.2.

Let
$\gamma\in{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
be any Legendrian embedding. Then, the natural inclusion

$i:{\mathfrak{Leg}}_{N,jN}^{\gamma}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\hookrightarrow{\mathfrak{Emb}}_{N,jN}^{\gamma}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$

is a homotopy equivalence.

As a consequence of this result there is not need to use any kind of geometric
invariant to study parametric families of connected Legendrians in the
$3$–sphere. Note, that the situation is maybe different in the case of
Lagrangian cobordisms/concordances/fillings [Cha10, EHK16, CG20, CDRGG15,
Pan17] where our methods do not apply. For instance the exact lagrangian
cobordisms defined by the loops of torus knots introduced in [K0́5] are not
distinguished by smooth invariants it remains to check whether formal
invariants can not distinguish them either [FMP17].

Observe that the previous result does not hold for the inclusion
${\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\hookrightarrow{\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$
because the inclusion $\operatorname{U}(2)\hookrightarrow\operatorname{SO}(4)$
is not an homotopy injection [FMP2].

According to Hatcher [Hat, Bud10], the group $\operatorname{Diff}(C_{\gamma})$
is homotopically discrete. In other words, each path-connected component of
${\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ is a $K(G,1)$.
Thus,

######  Theorem 1.0.3.

Let
$\gamma\in{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
be any Legendrian embedding. Then, the space
${\mathfrak{Leg}}_{N,jN}^{\gamma}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
is a $K(G,1)$. Equivalently, the path connected component
${\mathfrak{Leg}}^{\gamma}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ of the
space of Legendrian embeddings containing $\gamma$ is homotopy equivalent to
$\operatorname{U}(2)\times K(G,1)$.

This should be understood as a rigidity result in line with the Thurston-
Benequinn inequality; in fact, much more restrictive than this one.

It is an straightforward corollary the computation of the homotopy type of any
component of the space of Legendrian unknots and positive torus knots (See
Theorems 6.5.1 and 6.6.2).

The main result of this paper is the following statement that provides a quite
complete understanding of the topology of the group of contactomorphisms in
terms of the group of diffeomorphisms and the group of formal
contactomorphisms of a general tight contact $3$–fold. We say that a
diffeomorphism $\varphi$ admits a formal contactomorphism structure if it
comes equipped with a path of formal derivatives
$F_{s}\in\operatorname{Iso}(TM,\varphi^{*}TM)$, $s\in[0,1]$, starting at
$d\varphi=F_{0}$ and such that
$(F_{1})_{|\xi}\in\operatorname{Iso}(\xi,\varphi^{*}\xi)$. The pair
$(\varphi,F_{S})$ is said to be a formal contactomorphism. The space of formal
contactomorphisms is denoted by $\operatorname{FCont}(M,\xi)$. There is
natural inclusion

(5)
$\begin{array}[]{rccl}i_{F}:\operatorname{Cont}(M,\xi)&\hookrightarrow&\operatorname{FCont}(M,\xi)\\\
\varphi&\longmapsto&(\varphi,F_{s}\equiv d\varphi).\end{array}$

Fix any point $p\in M$. Denote by
$\operatorname{Cont}(M,\xi;\operatorname{rel}p)$ the group of
contactomorphisms that are the identity over ${\mathcal{O}p}(p)$ and by

$\operatorname{FCont}(M,\xi;\operatorname{rel}p)=\\{(\varphi,F_{s})\in\operatorname{FCont}(M,\xi):(\varphi,F_{s})_{|{\mathcal{O}p}(p)}={\operatorname{Id}}_{|{\mathcal{O}p}(p)}\\}$

the corresponding formal space. Do note that the homotopy type of the spaces
defined above is independent of the choice of $p$ since the space of
contactomorphisms acts transitively over $M$.

Consider the forgetful map

$\operatorname{pr}:\operatorname{FCont}(M,\xi;\operatorname{rel}p)\rightarrow\operatorname{Diff}(M;\operatorname{rel}p),$

which is actually a Serre fibration. Denote by

$D_{k}={\operatorname{Image}}(\pi_{k}(\operatorname{pr}))<\pi_{k}(\operatorname{Diff}(M;\operatorname{rel}p))$

the image of the induced homomorphisms on $\pi_{k}$.

Our main result is the following

######  Theorem 1.0.4.

Let $(M,\xi)$ be any compact tight contact $3$–manifold. Consider the
inclusion

$i_{D}:\operatorname{Cont}(M,\xi;\operatorname{rel}p)\hookrightarrow\operatorname{Diff}(M;\operatorname{rel}p).$

The following holds:

  1. (1)

The homomorphisms $\pi_{k}(i_{D})$ are injective for any $k\geq 1$.

  2. (2)

The inclusion
$i_{F}:\operatorname{Cont}(M,\xi;\operatorname{rel}p)\hookrightarrow\operatorname{FCont}(M,\xi;\operatorname{rel}p)$
is injective at the level of path–components.

  3. (3)

The image of $\pi_{k}(i_{D})$ is precisely $D_{k}$ for any $k$.

There has been a lot effort to compute the topology of the space of
contactomorphisms of a tight $3$-fold. Some examples: for the $\pi_{0}$, there
is a stream of articles by Giroux, Colin, Honda, Kanda, etc; in which they
provide a coarse classification of the space of contact structures [CGH09,
Col97, Gir00, Gir01b, Gir01a, Yut97, Hon00a, Hon00b]. In some of these
articles, they are able to deduce as a partial corollary the group
$\pi_{0}(\operatorname{Cont}(M,\xi))$

Some computations of high dimensional homotopy groups (either of contact
structures or contactomorphism groups) have been performed [Bou06, DG10, GM17,
GGP04, GK14]. However, it is fairly clear that there was no general method to
compute homotopy groups beyond the fundamental group either of the contact
structure space or the contactomorphism group. This makes Theorem 1.0.4 a
starting point for a whole bundle of classification results.

It is important to note that the previous result is based on Eliashberg’s
contractibility result about the space of tight contact structures on the
$3$-disks and Giroux convex surface theory. This is the starting point of the
argument in the proof.

The main difference between the specialized Theorem 1.0.1 and the general
Theorem 1.0.4 is the fact that for the particular case of knots complements,
we are able to show that $D_{K}=\pi_{k}(\operatorname{Diff}(M))$. It will
suffice to say that this is very specific of complements of Legendrian knots
in ${\mathbb{S}}^{3}.$ For the impatient reader, let us announce that it is
key that the knot is connected. In fact, this makes the biggest difference.
The study of Legendrian links possibly gives rise to exotic behaviours [CG20].
Secondly, it also helps that the diffemorphism group of the complementary of a
knot is homotopically discrete, this is far from true in the case of links.
Moreover, the proof explains in detail what are the important conditions to
improve into a global homotopy equivalence. They are summarized in Corollary
4.4.2.

Our main result provides an straightforward algorithm to compute the homotopy
type of the contactomorphism group of any compact tight $3$-fold with non-
empty boundary, assuming the knowledge of the homotopy type of the
diffeomorphism group. First consider the following fibration:

$\operatorname{Cont}(M,\xi;\operatorname{rel}p)\xrightarrow{i_{D}}\operatorname{Diff}(M;\operatorname{rel}p)\to{\mathfrak{C}Str}(M,\xi;\operatorname{rel}p),$

and we know that the map $\pi_{k}(i_{D})$ is injective for $k>0$. Moreover, we
know its image that is $D_{k}$. Thus, we have that
$\pi_{k}({\mathfrak{C}Str}(M,\xi;\operatorname{rel}p))$ equals
$\pi_{k}(\operatorname{Diff}(M;\operatorname{rel}p))/D_{k}=\pi_{k}(\operatorname{Diff}(M;\operatorname{rel}p))/\pi_{k}(\operatorname{Cont}(M,\xi;\operatorname{rel}p))$
for $k>1$. We will show several examples in Section 6.

As for the closed case, we have the following commutative grill of fibrations

$\textstyle{\operatorname{CFr}(M,\xi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fr}^{+}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}{\mathbb{P}}^{1}}$$\textstyle{\operatorname{Cont}(M,\xi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diff}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{C}Str}(M,\xi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cont}(M,\xi;\operatorname{rel}p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diff}(M;\operatorname{rel}p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{C}Str}(M,\xi;\operatorname{rel}p),\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 1. The grill diagram.

where $\operatorname{CFr}(M,\xi)$ is the total space of the bundle of contact
frames over $(M,\xi)$ and $\operatorname{Fr}^{+}(M)$ the bundle of positive
framings over $M$. Since every $3$-fold is paralellizable
$\operatorname{Fr}^{+}(M)=M\times\operatorname{SO}(3)$. If the first Chern
class $c_{1}(\xi)=0$ then it is also possible to identity
$\operatorname{CFr}(M,\xi)=M\times\operatorname{U}(1)$, this will be the case
in most of our examples.

There are several arrow chasing algorithms in order to recover the homotopy
type of the contactomorphism group if the homotopy type of the diffeomophism
group is known. A possible way (among others):

  * •

Compute $\operatorname{Diff}(M;\operatorname{rel}p)$ by studying the second
vertical fibration.

  * •

Then compute $\operatorname{Cont}(M,\xi;\operatorname{rel}p)$ by studying the
third horizontal fibration, i.e. use Theorem 1.0.4.

  * •

Compute $\operatorname{Cont}(M,\xi)$, by studying the first vertical
fibration.

What this long string of results hints is that $3$ dimensional contact
topology is mainly differential topology. In fact, there are (only) two
geometric controls on the whole process:

  * •

The lack of injectivity at $\pi_{0}$-level of the inclusion group of
contactomorphisms into the group of diffeomorphisms. That is shaping the
results about the space $\pi_{j}({\mathfrak{C}Str}(M,\xi))$ with $j=0,1$. The
place in which life has remained till now.

  * •

The understanding of the action of the contactomorphism group in the space of
Darboux balls. This measures the relation between the relative groups and the
absolute ones.

In a sense the rest of this article is pure algebraic topology. But, as Klaus
Niederkrüger once pointed out to us: “this does not mean that is easier at
all”. However, the reduction to Algebraic Topology is not completely
unexpected: it looks reasonable that $3$–dimensional contact topology looks
more topological than the higher dimensional one. What is unexpected is to
show that algebraic topology provides a lot of rigidity. This should not be a
surprise, the classification of contactomorphisms in the $3$-ball is an
extremely rigid result and it is the lever that moves all this article.
However, we keep forgetting how rigid it is and so, at least in the case of
the authors, we were expecting to find a bunch of exotic homotopy classes of
formal contactomophisms being represented by contactomorphisms. In that sense
(though clear from Eliashberg’s result in the $3$–ball): statement (1) from
Theorem 1.0.4 took us completely by surprise.

###  1.1. Outline.

Let us sketch the flow of the proofs. As we announced the key result is
Theorem 1.0.4. In order to get ready for the proof, first we collect several
tools in Section 2: a quick summary of convex surface theory, several results
about fibrations in contact topology (our spaces will typically be infinite
dimensional manifolds, so we must be careful with the definitions), the
relation between long (Legendrian/smooth) embeddings and (Legendrian/smooth)
embeddings, the equivalent ideas behind Theorem 1.0.4 in the smooth case as
developed by Hatcher. Finally an useful algebraic topology lemma, Lemma 2.6.2
will be stated.

Then Section 3 explains, and in some cases proves, several classical results
of $3$–dimensional Contact Topology in a multiparametric setup. This Section
is the most technical part of the article. In particular:

  * •

the contractibility of the tight contact structures in ${\mathbb{D}}^{3}$ with
fixed boundary extension, see Theorem 3.1.2, that is the starting lever of the
article;

  * •

the contractibility of the contactomorphism group of the ball relative to the
boundary, see Corollary 3.2.2;

  * •

Giroux $C^{0}$–density result about realizability of convex surfaces with a
chosen characterististic foliation. We will focus on the case of convex disks
whose boundary is a Lengedrian unknot with $\operatorname{tb}=-1$. And we will
prove that under some mild assumptions the space of standard convex disks is
proven to be homotopy equivalent to the space of smooth disks with the same
boundary. See Theorem 3.3.2.

  * •

Next, we offer the analogous proof to understand, at a homotopy level, the
inclusion of the space of standard convex embeddings of spheres on a tight
contact manifold into the space of smooth embeddings of spheres.

Let us digest the logic behind Theorem 1.0.4

  1. (1)

Fix a nice cell–decomposition of the contact $3$–fold (see Definition 4.1.1).
Denote by $G_{j}$ the $j$–skeleton and $G_{-1}=\emptyset$.

  2. (2)

Our dream is to prove that if an element
$\beta\in\pi_{k}(\operatorname{Cont}(M^{3},\xi;\operatorname{rel}p))$
satisfies that $i_{D}(\beta)=0$ then $\beta=0$ and moreover for an element
$\beta\in\pi_{k}(\operatorname{Diff}(M;\operatorname{rel}p))$ such that it
admits a formal contact structure, i.e.
$\beta=\pi_{k}(\operatorname{pr})(\delta)$ for some
$\delta\in\pi_{k}(\operatorname{FCont}(M,\xi;\operatorname{rel}p)$, then you
can deform it through a smooth family (not necessarily formal) to obtain a new
family that is contact. This is what we advertise in the statement. However,
we will find obstructions on the process, e.g. [Bou06, DG10, Gir01b, GM17];

  3. (3)

The logic game that we apply is to work by steps decreasing the dimension of
the cells $j=3,2,1,0$.

  4. (4)

The rank $j$ case can be stated as follows. We have as induction hypothesis
that the map

$\pi_{k}(\operatorname{Cont}(M^{3},\xi;\operatorname{rel}G_{j}))\to\pi_{k}(\operatorname{Diff}(M^{3};\operatorname{rel}G_{j}))$

is injective and moreover any element in
$\pi_{k}(\operatorname{Diff}(M^{3};\operatorname{rel}G_{j}))$ that admits a
formal contact structure, can be deformed through diffeomorphisms into an
element in $\pi_{k}(\operatorname{Cont}(M^{3},\xi;\operatorname{rel}G_{j}))$.
The required statement is that we want to prove that is true for $j-1$, i.e.

$\pi_{k}(\operatorname{Cont}(M^{3},\xi;\operatorname{rel}G_{j-1}))$

should be mapped injectively into

$\pi_{k}(\operatorname{Diff}(M^{3},\xi;\operatorname{rel}G_{j-1}))$

and given an element in

$\beta\in\pi_{k}(\operatorname{Diff}(M^{3};\operatorname{rel}G_{j-1})),$

that lies in the image of

$\pi_{k}(\operatorname{pr}):\pi_{k}(\operatorname{FCont}(M^{3},\xi;\operatorname{rel}G_{j-1}))\to\pi_{k}(\operatorname{Diff}(M^{3};\operatorname{rel}G_{j-1})),$

there is an element in

$\pi_{k}(\operatorname{Cont}(M^{3},\xi;\operatorname{rel}G_{j-1}))$

whose smooth class is given by $\beta$.

  5. (5)

In the first step, $j=3$, realize that the induction hypothesis is void. This
is because $G_{3}=M^{3}$ and that the group conformed by the identity element
obviously satisfies the induction hypothesis. Actually, what we prove for
$j-1=2$ is stronger than required. We are assuming that away from a
neighborhood of the $2$-skeleton (a finite union of Darboux balls) our maps
are the identity, to conclude it is enough to realize that contactomorphisms
relative to the boundary of the ball are homotopically equivalent to
diffeomorphisms relative to the boundary of the ball. This is Corollary 3.2.2
of our lever Theorem 3.1.2.

  6. (6)

In the second step, $j=2$ we assume that our families are the identity in the
boundary of the $1$-skeleton, for instance for injectivity: start with a
family $\varphi^{z,1}\in\operatorname{Cont}(M^{3};\operatorname{rel}G_{1})$,
$(z,1)\in\partial{\mathbb{D}}^{k}={\mathbb{S}}^{k-1}$, such that they extend
to a family
$\varphi^{z,r}\in\operatorname{Diff}(M^{3};\operatorname{rel}G_{1})$, for
$(z,r)\in{\mathbb{D}}^{k}$. We start with the family $\varphi^{z,1}$ that by
induction hipothesis restrict to the identity on a neighborhood of the
$1$-skeleton. Given a $2$-cell $c_{2}$, our goal is to show that the family of
contactomorphism leave it invariant. The rules of the game are the following:
we are allowed to deform our family $\varphi^{z,1}$ by post-composing with a
family of contact flows in order to push back the family
$\varphi^{z,1}(c_{2})$ to the original $2$-cell $c_{2}$. It will be
instrumental for that the smooth capping by diffeomorphisms that will provide
an isotopy through smooth disks from the images to the original one. At this
point, the cross of the argument develops: we need to approximate the
multiparametric family of smooth disks into a family of convex disks that by
extension of isotopy provide a family of flows that pushes back the convex
disks to the initial one. For that, we will need a multiparametric version of
convex surface theory, see Subsection 3.3. A similar argument provides the
surjection.

  7. (7)

The first two steps $j=3,2$ are unobstructed, even considering the inclusion
$\operatorname{Cont}(M,\xi)\to\operatorname{Diff}(M)$. This is the content of
Theorem 4.2.1, that is of independent interest. In fact, this result is enough
to prove that the space of long Legendrian unknots is contractible, something
needed to actually complete the third step $j=1$.

  8. (8)

Let us focus on the step $j=1$. In this case, it is simple to reduce the
problem to the fact that the space of long Legendrian unknots is contractible
(in other words homotopically equivalent to the space of long smooth unknots).
This allows us to show that the family of contactomorphisms can be deformed in
such a way that it preserves the $1$-edges of the $CW$-decomposition. In fact,
this just works for a family of diffeomorphisms (no need for them to be
formal). Once, this is done, one may be tempted to think that we have solved
the problem. However, we need to recall that we should preserve a neighborhood
of the edge, and that is equivalent to say that we preserve a framing of the
normal (this implies that is the identity at first order). But clearly, this
is not always the case for general diffeomorphims: for instancce it may happen
that our arc with local coordinates
$(z,r,\theta)\in[0,1]\times{\mathbb{D}}^{2}$ has the following local behaviour
$(z,r,\theta)\to(z,r,\theta+2\pi kz)$, with
$k\in{\mathbb{Z}}\backslash\\{0\\}$. One may be tempted to think that this
cannot happen in families, but it is still possible. Imagine that you have a
family of diffeomorhpisms $\phi^{z,1}$ that act trivially over an arc
preserving it and its normal bundle. It is very simple to produce a family of
diffeomorphisms $\phi^{z,r}$; that parametrizes $(z,r)\in{\mathbb{B}}^{k}$
such that $\phi^{0,0}$ is a diffeomorphism that preserves the arc and,
however, rotates the normal bundle. This can happen as soon as the family
$\phi^{z,r}$ does not preserve the arc for $r\in(0,1)$.

A way out is to assume that the map is formal contact (being precise at
$1$-skeleton level, but we just simplify the statement of the result saying
that they are formal contactomorphisms.) If that is the case, it is fairly
clear that the image of the contact framing on the normal bundle over $\gamma$
is homotopic to the contact framing associated to the image, if our
deformation is through formal contactomorphisms we preserve the framing and we
are done. In the injective case, that is trivially achieved because the
deformation is actually through contactomomrphims. However, in the surjective
case we need a formal path to follow and this is the reason of the technical
statement in the Theorem 1.0.4.

  9. (9)

Let us focus on the last step $j=0$. In particular, the strategy as it is
clear from the statement of Theorem 1.0.4 is just to give up and assume that
you have sorted it out by hypothesis. Why do not we assume that if there is a
formal contact isotopy to the identity, then we just deform the $0$-cells back
to the initial position? The problem is explained in Subsection 4.5: a
contactomorphism that is homotopic to the identity as formal contactomorphism
is shown to not satisfy the injectivity asumption. However, we will check that
there is not CW-decomposition such that if you deform the images of the
$0$-skeleton in order to make sure that the contactomorphism is the identity
over a small neighborhood of the $0$-skeleton, then the new contactomorphism
does not remain formallly isotopic to the identity relative to the $0$-cells.
This is a classical fact in obstruction theory: secondary obstructions may
appear if you have an unobstructed problem and try to solve it by induction in
the skeleton. So, we implement a radical solution, we assume that the zero
skeleton is sorted out and we start in the $1$-skeleton.

The careful reader may realize at this point that the technically challenging
steps are $j=1$ and $2$. For $j=2$, we are just trying to push a family of
disks that are the image of a $2$-cell of the Legendrian skeleton back to this
disk. What we need to make it work is a multiparametric convex theory, at
least for the standard convex disk with boundary the Legendrian unknot. To
prove this is the content of Subsection 3.3, though in a sense the whole
Section 3 is a preparation for that result.

Lastly, the key result to attack the case $j=1$ is sorted out as a Corollary
of the discussion for $j=2$, that is Corollary 4.2.2.

Section 5 contains all the proofs of the Theorems in the Introduction as
simple consequences of Theorem 1.0.4. The most relevant one is Theorem 1.0.1,
in which by using the specific topology of $C_{\gamma}$ we actually check that
any family of diffeomorphisms of it can be given a formal contact at
$1$-skeleton level, enough to remove the hypothesis of formality and state the
result.

Section 6 shows several easy applications of the article:

  * •

We use Theorem 1.0.4 to explicitly compute the homotopy type of the group
$\operatorname{Cont}(M,\xi)$ in some examples. See Subsection 6.1.

  * •

We deduce the analogous of Theorem 1.0.3 for transverse embeddings (see
Theorem 6.2.1).

  * •

We show that the scanning map
$i:{\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\hookrightarrow{\mathfrak{FLeg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
induces an injective homomorphism at $\pi_{k}$, $k>0$, that is non-surjective.
This lack of surjectivity can be thought as a generalized Thurston-Bennequin
innequality. See Theorem 6.3.2.

  * •

We prove a relative version of Eliashberg classification result of tight
contact structures over ${\mathbb{S}}^{3}$ for the complements of formally
simple Legendrians. See Theorem 6.4.6.

  * •

We compute the homotopy type of the space of unknotted Legendrian embeddings
and parametrized positive Legendrian torus knots. See Theorems 6.5.1 and
6.6.2.

  * •

We compute the homotopy type of the space of parametrized Legendrian
$(n,n)$-torus links with maximal $\operatorname{tb}$. See Theorem 6.7.1.

  * •

We deduce similar statements for Legendrians embeddings in the standard
contact ${\mathbb{R}}^{3}$. See Theorems 6.8.2 and 6.8.3.

  * •

We describe the homotopy groups of the space of embedded Legendrians in
$({\mathbb{S}}^{3},\xi_{\operatorname{std}})$. See Theorem 6.9.3.

  * •

We deduce a nice application about the representability of different tight
${\mathbb{R}}^{3}$’s in terms of a parametric identification with the fixed
standard one, Theorem 6.10.3.

###  1.2. Acknowledgements.

Special thanks to Ángel González for the uncountable number of hours that he
has spent listening to us. We want to acknowledge the interest and kindness of
Ryan Budney that explained to us very basic things when we were getting into
the fascinating world of smooth embeddings theory. Moreover, he was a great
host when we visited him in Victoria where he taught us about sophisticated
homotopy computations and what is more important about crab fishing and
digesting. First author wants to thank Roger Casals, Yasha Eliashberg and
Viktor Ginzburg for hosting him and asking him all the right questions in a
visit to California in which this whole project unexpectedly began to take
shape; and to Fabio Gironella for useful conversations/explanations about $3D$
contact topology. The second author wants to thank Álvaro del Pino for his
support and interest in this project. Also he would like to thank the Geometry
group at Utrecht University where he was given a nice environment to develop
this and other projects. The authors want to acknowledge the support of the
Spanish national grant with reference number PID2019-108936GB-C21
(MINECO/FEDER) and by the excellence project CEX2019-000904-S. The first
author is supported by Beca de Personal Investigador en Formación UCM. The
second author is funded by Programa Predoctoral de Formación de Personal
Investigador No Doctor del Departamento de Educación del Gobierno Vasco.

##  2\. Preliminaries.

This Section reviews several quite standard results in Contact Topology.
Though, they are very standard for the expert in each subarea, it may happen
that even the usual reader of Contact Topology is not familiar with some of
them. As for the last two Subsections, they are focused in a number of
standard results of $3$-dimensional differential topology. Specially
remarkable for us is Lemma 2.6.2 that is used in a continuous way all over the
article. It is a frequent tool in algebraic topology and probably less
frequent in contact topology.

###  2.1. Convex surface theory in tight contact $3$–manifolds.

We will recall some facts about Convex Surface Theory that we will need in the
article. The reader is referred to [Hon00a, HH19, Gir91, Mas14] for further
details.

Let $(M,\xi)$ be a contact $3$-manifold. A properly embedded surface
$S\subseteq M$ is said to be convex if there exists a contact vector field $X$
that is transverse to $S$. A convex embedding of $S$ into $(M,\xi)$ is any
embedding $e:S\rightarrow M$ such that $e(S)\subseteq M$ is convex. The
$1$–dimensional singular foliation $e^{*}\xi$ on $S$ is called the
characteristic foliation of the surface. A convex surface $S$ has a
neighbourhood in $M$ contactomorphic to
$(S\times{\mathbb{R}},\ker(fdt+\beta))$ where $t$ is the
${\mathbb{R}}$–coordinate, $\beta\in\Omega^{1}(S)$ is the pullback, via the
inclusion, of some contact form $\alpha$ of $(M,\xi)$ to $S$, and $f\in
C^{\infty}(S)$. The zero set of $f$ defines an embedded $1$–dimensional
submanifold of $\Gamma\subseteq S$ that is called the dividing set. This
construction depends on the choice of the contact vector field $X$, but since
the space of contact vector fields transverse to $S$ is contractible the
isotopy class of $\Gamma$ is well–defined. We will write $\Gamma_{X}$ instead
of $\Gamma$ whenever we want to make clear the choice of contact vector field
that we are using.

######  Theorem 2.1.1 (Giroux Approximation Theorem [Gir01b, Hon00a]).

Let $(M,\xi)$ be any contact $3$–manifold. Let $S\subseteq M$ be a compact
surface. If $S$ has non-empty boundary we assume that is Legendrian and that
the twisting number between the contact framing of the normal framing of
$\partial S$ and the induced by $S$ is negative. Then,

  * •

If $\partial S=\emptyset$ there exists a $C^{\infty}$ perturbation of $S$ that
makes it convex.

  * •

If $\partial S\neq\emptyset$ there exists a pertubation, fixed at the
boundary, which is $C^{0}$ near the boundary and $C^{\infty}$ in the interior,
of $S$ that makes it convex.

We will apply the previous Theorem to families of disks which share a small
neighborhood of the boundary, so we will be assuming that they will be convex
near the boundary. This allows us to assume that the perturbation is
$C^{\infty}$ small in those applications.

######  Theorem 2.1.2 (Giroux Tightness Criterion [Gir01a]).

Let $S\subseteq(M,\xi)$ be a convex surface in a tight contact $3$–manifold.
Then,

  * •

if $S$ is an sphere the dividing set is connected,

  * •

if $S$ is not an sphere the dividing set does not contain any homotopically
trivial curve.

Let $e:S\rightarrow(M,\xi)$ be a convex embedding. Denote the characteristic
foliation of $S$ by $\mathcal{F}=e^{*}\xi$. Denote by
${\mathfrak{Emb}}(S,(M,\xi),\mathcal{F})$ the space of embeddings
$j:S\rightarrow M$ such that $j^{*}\xi=\mathcal{F}$. Fix a contact vector
field $X$ transverse to $e(S)$ and let $\Gamma_{X}$ be the dividing set
defined by $X$. Denote by ${\mathfrak{Emb}}(S,(M,\xi),\Gamma_{X})$ the space
of embeddings $j:S\rightarrow M$ such that $j^{*}\xi$ is divided by
$\Gamma_{X}$. In the case that $\partial S\neq\emptyset$ we assume that all
the embeddings coincide in an open neighbourhood of the boundary and that the
boundary is Legendrian.

######  Theorem 2.1.3 (Giroux Realization Theorem [Gir01b]).

The natural inclusion

${\mathfrak{Emb}}(S,(M,\xi),\mathcal{F})\hookrightarrow{\mathfrak{Emb}}(S,(M,\xi),\Gamma_{X})$

is a homotopy equivalence.

###  2.2. Fibrations in contact topology.

The following lemma is well known. A good source for most of the fibrations
that we are going to describe are the P. Massot notes [Mas15] which are a
detailed version of the results explained in [GM17]. The reader is also
referred to [Gei08] where the non-parametric version of these results are also
proven. Given a contact manifold $(M,\xi)$ we will denote by
${\mathfrak{C}Str}(M;\xi)$ the space of contact structures over $M$ which are
isotopic to $\xi$.

######  Lemma 2.2.1.

Let $(M,\xi)$ be a compact contact $3$–manifold. Fix a Legendrian embedding
$\gamma\in{\mathfrak{Leg}}(M,\xi)$. Consider also $j:S\rightarrow M$ an
embedding of a surface $S$ into $M$ and denote by $\mathcal{F}=j^{*}\xi$ the
characteristic foliation of $S$. We denote by
${\mathfrak{Emb}}(S,M;\mathcal{F})$ the space of embeddings of $e:S\rightarrow
M$ such that $e^{*}\xi=\mathcal{F}$.

  * •

The orbit map

$\operatorname{Cont}(M,\xi)\rightarrow{\mathfrak{Leg}}(M,\xi),\varphi\mapsto\varphi\circ\gamma;$

is a fibration with fiber
$\operatorname{Cont}(M,\xi;\operatorname{rel}\gamma).$ The same holds for
transverse embeddings with the only difference that the fiber is the group of
contactomorphisms that fix the transverse embedding and not an open
neighbourhood of it.

  * •

The orbit map

$\operatorname{Diff}(M)\rightarrow{\mathfrak{C}Str}(M,\xi),\varphi\mapsto\varphi_{*}\xi;$

is a fibration with fiber $\operatorname{Cont}(M,\xi).$

  * •

The orbit map

$\operatorname{Cont}(M,\xi)\rightarrow{\mathfrak{Emb}}(S,M;\mathcal{F}),\varphi\mapsto\varphi\circ
j;$

is a fibration.

###### Proof.

All these results are correctly proven in [GM17]. We just recall how to prove
the last one. In particular, the homotopy lifting property of the fibration

$\operatorname{Cont}(M,\xi)\rightarrow{\mathfrak{Emb}}(S,M;\mathcal{F}).$

Take a disk

$j^{z,r}:{\mathbb{D}}^{z}\to{\mathfrak{Emb}}(S,M;\mathcal{F}),(z,r)\in{\mathbb{D}}^{k}.$

Apply the (smooth) isotopy extension theorem to the family of surfaces
$j^{z,r}$, where $r\in[0,1]$ and $z\in{\mathbb{S}}^{k-1}$ is a fixed. This
produces a family of diffemorphisms $\varphi^{z,r}$ such that
$\varphi^{z,r}\circ j^{z,0}=j^{z,r}$ and $\varphi^{z,0}={\operatorname{Id}}$.
Moreover, construct the family $\varphi^{z,r}$ in such a way that the push-
forward by $\varphi^{z,r}$ of the contact structure evaluated over
$j^{z,0}(S)$ coincides with the contact structure evaluated over the image of
$j^{z,r}$. This can be done because the contact structure in a neighbourhood
of a surface is determined by the characteristic foliation. Thus,
$\xi_{z,r}=(\varphi_{z,r})_{*}\xi$ is a family of contact structures
$\xi_{z,r}$ that coincides with $\xi$ over $j^{z,r}(S)$. Apply Gray’s stabiliy
to produce a family of diffeomorphism $G^{z,r}$ satisfying that
$(G^{z,r})_{*}\xi_{z,r}=\xi$. By construction $G^{z,r}$ is the identity over
$j^{z,r}$. Thus,

$G^{z,r}\circ\varphi^{z,r}\in\operatorname{Cont}(M,\xi)$

is a disk of contactomorphisms of $(M,\xi)$ satisfying that

$G^{z,r}\circ\varphi^{z,r}\circ j^{z,0}=G^{z,r}\circ j^{z,r}=j^{z,r}.$

This completes the proof. ∎

###  2.3. The space of Darboux balls in a contact manifold.

Alexander’s trick allows to prove that the space
${\mathfrak{Emb}}^{+}({\mathbb{D}}^{n},{\mathbb{R}}^{n})$ of orientation
preserving embeddings ${\mathbb{D}}^{n}\rightarrow{\mathbb{R}}^{n}$ linearise,
i.e. it is homotopy equivalent to $\operatorname{GL}^{+}(n,{\mathbb{R}})$.
This, together with the Isotopy Extension Theorem, implies that on a closed
oriented $n$–manifold $N^{n}$ the space
${\mathfrak{Emb}}^{+}({\mathbb{D}}^{n},N^{n})$ is homotopy equivalente to the
total space of the oriented frame bundle $\operatorname{Fr}^{+}(N)$.
Explicitly, there is a map of fibrations

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0.0pt\hbox{$\textstyle{N^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
41.50488pt\raise-103.2055pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{{\operatorname{Id}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
79.18329pt\raise-108.63606pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 10.0pt\raise
0.0pt\hbox{$\textstyle{N^{n}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Where the maps between the fibers and between the bases are homotopy
equivalences. Thus, the natural map
${\mathfrak{Emb}}^{+}({\mathbb{D}}^{n},N^{n})\rightarrow\operatorname{Fr}^{+}(N^{n})$
is a homotopy equivalence.

In the contact category there is also an Alexander trick. Indeed, in
$({\mathbb{R}}^{2n+1},\xi_{\operatorname{std}}=\ker(dz-\sum_{i}y_{i}dx_{i}))$
the dilation

$\delta_{t}:{\mathbb{R}}^{2n+1}\rightarrow{\mathbb{R}}^{2n+1},(\textbf{x},\textbf{y},z)\mapsto(t\textbf{x},t\textbf{y},t^{2}z);$

is a contactomorphism for any $t>0$. This implies that the space

${\mathfrak{CEmb}}(({\mathbb{D}}^{2n+1},\xi_{\operatorname{std}}),({\mathbb{R}}^{2n+1},\xi_{\operatorname{std}}))$

of co–oriented embeddings of Darboux balls
$({\mathbb{D}}^{2n+1},\xi_{\operatorname{std}})$ into
$({\mathbb{R}}^{2n+1},\xi_{\operatorname{std}})$ is homotopy equivalent to the
space of contact framings, i.e. to $\operatorname{U}(n)$111Being precise one
should write the positive conformal symplectic group. This group is homotopy
equivalent to $\operatorname{U}(n)$.. We refer the reader to [Gei08, Section
2.6.2] for further details. In particular, the Isotopy Extension Theorems in
Contact Topology implies, in the same way as in the smooth case, that

######  Lemma 2.3.1.

Let $(N,\xi)$ be a closed co–oriented $(2n+1)$–contact manifold. The space
${\mathfrak{CEmb}}(({\mathbb{D}}^{2n+1},\xi_{\operatorname{std}}),(N,\xi))$ is
homotopy equivalent to the total space of the bundle of contact framings
$\operatorname{CFr}(N,\xi)$ over $(N,\xi)$, which has fiber
$\operatorname{U}(n)$; i.e. a Darboux ball is determined by the centre of the
ball and the induced framing of $\xi$ at that point.

In particular, for $({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ the space of
parametrized Darboux balls is homotopy equivalent to
${\mathbb{S}}^{3}\times\operatorname{U}(1)$.

###  2.4. Legendrian embeddings in
$({\mathbb{S}}^{3},\xi_{\operatorname{std}})$.

Let $({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ be the standard tight
contact $3$–sphere, i.e. $\xi_{\operatorname{std}}$ is defined as the complex
tangencies of $T{\mathbb{S}}^{3}$ with respect to the standard complex
structure in ${\mathbb{C}}^{2}\supseteq{\mathbb{S}}^{3}$.

######  Definition 2.4.1.

Let $(M,\xi)$ be a contact $3$–manifold. An embedded oriented circle
$L\subseteq M$ is said to be Legendrian if it is everywhere tangent to $\xi$.
A Legendrian embedding is any embedding of the circle into $M$ that
parametrizes a Legendrian.

Denote by $\widehat{{\mathfrak{Leg}}}(M,\xi)$ the space of oriented
Legendrians in $(M,\xi)$ and by ${\mathfrak{Leg}}(M,\xi)$ the space of
Legendrian embeddings. In this article we are mostly interested in the case of
$({\mathbb{S}}^{3},\xi_{\operatorname{std}})$.

Let
${\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})=\\{\gamma\in{\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}}):\gamma(0)=N,\gamma^{\prime}(0)=jN\\}$
be the space of long Legendrian embeddings. We will see later that this space
is homotopy equivalent with the space of long Legendrian embeddings in the
usual knot theorist sense: Legendrian embeddings into
$({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ that coincide in an open
neighbourhood of the north pole $N$ with the Legendrian great circle
$\gamma(t)=(\cos t,\sin t)\in{\mathbb{S}}^{3}\subseteq{\mathbb{C}}^{2}$. This
justifies the name of the space.

######  Lemma 2.4.2.

There exist a homotopy equivalence

(6)
$\Phi:{\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\rightarrow\operatorname{U}(2)\times{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}}).$

###### Proof.

We assume without loss of generality that all the Legendrian embeddings
$\gamma$ satisfy that $||\gamma^{\prime}(0)||=1$. By the Legendrian condition
we have a natural map
${\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\rightarrow\operatorname{U}(2),\gamma\mapsto
A_{\gamma}=(\gamma(0)|\gamma^{\prime}(0)).$ The homotopy equivalence is given
by

(7)
$\begin{array}[]{rccl}\Phi:{\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})&\longrightarrow&\operatorname{U}(2)\times{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\\\
\gamma&\mapsto&(A_{\gamma},A_{\gamma}^{-1}\gamma).\end{array}$

∎

######  Remark 2.4.3.

Let ${\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ be the space of
smooth embeddings of the circle into ${\mathbb{S}}^{3}$. Denote also by
${\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})=\\{\gamma\in{\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{S}}^{3}):\gamma(0)=N,\gamma^{\prime}(0)=jN\\}$
the subspace of smooth long embeddings. The previous result is just the
Legendrian version of the classical fibration

$\begin{array}[]{rccl}{\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{S}}^{3})&\longrightarrow&V_{4,2}=\operatorname{SO}(4)/\operatorname{SO}(2)\\\
\gamma&\mapsto&(\gamma(0),\gamma^{\prime}(0)),\end{array}$

where the fiber over the point $(N,jN)\in V_{4,2}$ is precisely the space of
smooth long embeddings
${\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$. Observe that the
map
${\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\rightarrow\operatorname{U}(2),\gamma\mapsto
A_{\gamma}$; defined in the previous proof is just the restriction of this
fibration to the subspace
${\mathfrak{Leg}}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\subseteq{\mathfrak{Emb}}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$.

######  Definition 2.4.4.

Let $\gamma^{z}$, $z\in{\mathbb{S}}^{k}$, be any sphere of Legendrian
embeddings in $({\mathbb{S}}^{3},\xi_{\operatorname{std}})$. The rotation
class of the sphere is the homotopy class

$\operatorname{rot}_{\pi_{k}}(\gamma^{z})=[A_{\gamma^{z}}]\in\pi_{k}(\operatorname{U}(2)).$

One important fact that we are going to use is the following technical lemma,
that is essentially a consequence of the Tubular Neighbourhood Theorem for
Legendrians

######  Lemma 2.4.5.

Let $(M^{3},\xi)$ be any compact contact $3$–manifold. Let $L\subseteq M$ be
any embedded Legendrian circle. Then, the space of contactomorphisms of
$(M,\xi)$ that preserves $L$ (pointwise) is homotopy equivalent to the space
of contactomorphism that preserves a tubular neighbourhood of $L$ (resp.
pointwise).

###### Proof.

By the Weinstein neighborhood theorem the neighborhood of the embedding is
completely determined by the sole Legendrian embedding. Therefore, given a
contactomorphism of the Weinstein neighbourhood to the Weinstein neighbourhood
we can just check that it is determined up to $C^{1}$-perturbation by a
diffeomorphism of the image of the embedding. If we assume that we are fixing
the image of the embedding pointwise, this determines- up to homotopy- a
unique lift that is the $1$-jet extension of the identity diffeomorphism. ∎

####  2.4.1. Long Legendrian arcs.

######  Definition 2.4.6.

A long Legendrian arc in
$({\mathbb{B}}^{3}(x,y,z),\xi_{\operatorname{std}}=\ker(dz-ydx))$ is a
Legendrian embedding
$l:[-1,1]\rightarrow({\mathbb{B}}^{3},\xi_{\operatorname{std}})$ such that
$l_{{\mathcal{O}p}\\{-1,1\\}}=(e_{\operatorname{ref}})_{|{\mathcal{O}p}\\{-1,1\\}}$,
where $e_{\operatorname{ref}}(t)=(t,0,0)$.

Denote the space of long Legendrian arcs by
${\mathfrak{LongLeg}}({\mathbb{B}}^{3},\xi_{\operatorname{std}})$. There is a
natural inclusion

(8)
$\mathcal{T}:{\mathfrak{LongLeg}}({\mathbb{B}}^{3},\xi_{\operatorname{std}})\hookrightarrow{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}}),$

induced by a contact embedding
$\varphi:({\mathbb{B}}^{3},\xi_{\operatorname{std}})\hookrightarrow({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
which identifies the ball $({\mathbb{B}}^{3},\xi_{\operatorname{std}})$ with
the hemisphere $\\{(z_{1},z_{2})\in{\mathbb{S}}^{3}:(z_{1})<0\\}$ and sends
the Legendrian arc $e_{\operatorname{ref}}(t)=(t,0,0)$ to
$(\cos\frac{\pi(t+2)}{2},\sin\frac{\pi(t+2)}{2})$. In other words
$\varphi(-1,0,0)=(0,1)$ and $\varphi(1,0,0)=(0,-1)$ are in the dividing set
and in antipodal position for the convex sphere $\\{(z_{1})=0\\}$. The
inclusion is defined sending a long Legendrian arc $l$ to the concatenation
with the two trajectories of the radial vector field starting in $(0,-1)$ and
$(0,1)$. It is important to remark that this explicit construction works
because both points are in the dividing set and in antipodal position in the
characteritic foliation for the sphere $\\{(z_{1})=0\\}$ and the flow leaves
the characteristic foliation invariant and thus we get that the arc closes
smoothly.

The following Lemma justifies the notion of long Legendrian embedding that we
will use in this article.

######  Lemma 2.4.7.

The inclusion

$\mathcal{T}:{\mathfrak{LongLeg}}({\mathbb{B}}^{3},\xi_{\operatorname{std}})\hookrightarrow{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$

is a homotopy equivalence.

###### Proof.

We must build a homotopy inverse of $\mathcal{T}$. Consider a long Legendrian
embedding
$\tilde{\gamma}:{\mathbb{S}}^{1}\to({\mathbb{S}}^{3},\xi_{\operatorname{std}})$,
i.e. $\tilde{\gamma}(0)=N$ and $\tilde{\gamma}^{\prime}(0)=jN$.

We say that the long meridian knot $\tilde{\gamma}_{\operatorname{ref}}$ is
the compactification of the long arc $e_{\operatorname{ref}}(t)=(t,0,0)$. I.e.
the standard Legendrian unknot $\tilde{\gamma}_{\operatorname{ref}}(t)=(\cos
t,\sin t)$. Define the number

$\epsilon:{\mathfrak{Leg}}_{N,jN}({\mathbb{B}}^{3},\xi_{\operatorname{std}})\times{\mathbb{R}}\to{\mathbb{R}}^{+}$

as follows. For each point $p=\tilde{\gamma}(t)$ of the image of
$\tilde{\gamma}$ compute the maximum of the following two positive numbers,

  1. (1)

the shortest distance between the point and the point in the meridian long
unknot $\tilde{\gamma}_{\operatorname{ref}}(t)$,

  2. (2)

the angle of the derivative of the long knot with respect to the negative
radial vector field at the point $p=\tilde{\gamma}(t)$.

Find the element $p_{0}$, closest to the north pole over the image of
$\tilde{\gamma}$, for which $\epsilon(p)>\varepsilon$, for some universal
constant $\varepsilon>0$ sufficiently small.

Define $\mu(\tilde{\gamma})=\operatorname{dist}(N,p_{0})$, i.e the distance
between the north pole $N$ and $p_{0}$ with respect to the round metric. We
declare $\mu(\tilde{\gamma})=\pi$ if $p_{0}$ does not exist.

Define $\phi^{X}_{t}$ the contact flow generated by the minus gradient of the
height function of the $3$-sphere that is
${\operatorname{Image}}(z_{2})=y_{2}$, i.e. its trajectories are just the
meridians. For every $\tilde{\gamma}$, there is a $\tau_{\tilde{\gamma}}$ that
is the minimum time $\tau_{\tilde{\gamma}}$ for which
$\hat{\gamma}=\phi^{X}_{\tau_{\tilde{\gamma}}}\circ\tilde{\gamma}$ satisfies
that $\mu(\hat{\gamma})=\frac{\pi}{2}+\varepsilon$. Recall that the distance
between the north and the south pole is $\pi$. Moreover, in the case in which
$\mu(\tilde{\gamma})>\frac{\pi}{2}+\varepsilon$ declare
$\tau_{\tilde{\gamma}}=0$. We have that $\tau_{\tilde{\gamma}}$ is continuous
over the space of long smooth knots. Finally define the continuous
transformation

$\displaystyle{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
$\displaystyle\to$
$\displaystyle{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
$\displaystyle\tilde{\gamma}$ $\displaystyle\to$
$\displaystyle\hat{\gamma}=\phi^{X}_{\tau_{\tilde{\gamma}}}\circ\tilde{\gamma}$

Now, reparametrize $\hat{\gamma}$ in order to make sure that the intersection
with the equator in ${\mathbb{S}}^{3}$ happens at time $t=-1$. Recall that the
height function corresponds to the coordinate $y_{2}=im(z_{2})$. Moreover,
assume that all the Legendrian knots $\hat{\gamma}$ are reparametrized in such
a way that:

  * •

${\mathbb{S}}^{1}={\mathbb{R}}/(4{\mathbb{Z}})$,

  * •

$\hat{\gamma}(2)=N$

  * •

$\hat{\gamma}(0)=S$

  * •

$\hat{\gamma}\bigcap\\{y_{2}\geq 0\\}$

  * •

$y_{2}(\hat{\gamma}(t))=y_{2}(\tilde{\gamma}_{\operatorname{ref}}(t))$

It is clear that $\hat{\gamma}$ is $C^{1}$–close to
$\gamma_{\operatorname{ref}}$ over a $\varepsilon$ neighborhood of the
northern hemisphere, that is parametrized by $|t|>1-\varepsilon$. Fix a
Weinstein chart of $\gamma_{\operatorname{ref}}$ that is defined on the
intersection of the $\varepsilon$ regular neighborhood of
$\gamma_{\operatorname{ref}}$ and the $\varepsilon$-neighborhood of the
northern hemisphere $\tilde{H}=\\{y_{2}\geq-1+\varepsilon\\}$, make sure that
you choose a parametrization $\gamma^{0}_{|y_{2}\geq-\varepsilon}$ that runs
between $[1-\varepsilon,-1+\varepsilon]\subset{\mathbb{R}}/(4{\mathbb{Z}})$.
By using the fixed Weinstein chart, we can identify over $\tilde{H}$, the
Legendrian $\hat{\gamma}$ with the $1$-jet $j^{1}\hat{f}$ for some smooth
function. Choose a cut-off function $c:(-1+\varepsilon,1-\varepsilon]\to[0,1]$
such that:

  * •

$c(x)=1$, for $x\in[-1-\delta,1+\delta],\varepsilon>>\delta>0$,

  * •

$c(x)=0$, for $x\in[1,1+\varepsilon-\delta]\varepsilon]$

Now define $\hat{f}_{s}=c(t)(1-s)f(t)+(1-c(t))f(t)$ that coincides with $f$
near the ends and with the zero section over the northern hemisphere. The
associated legendrians are now, when restricted to the southern hemisphere,
elements of ${\mathfrak{LongLeg}}({\mathbb{B}}^{3},\xi_{\operatorname{std}})$,
so we have found a map $\mathcal{S}$, clearly continuous,

${\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\rightarrow{\mathfrak{LongLeg}}({\mathbb{B}}^{3},\xi_{\operatorname{std}}).$

The careful reader may check that $\mathcal{T}$ and $\mathcal{S}$ are homotopy
inverses. ∎

###  2.5. Smooth Case. Hatcher’s Theorems.

In this Section we recall some known facts about diffeomorphism groups of
$3$-folds and the topology of the space
${\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ of smooth long
embeddings into ${\mathbb{S}}^{3}$. The scheme that are we going to follow in
the Legendrian case is developed out of the ideas explained in this Section.

####  2.5.1. The Smale Conjecture and knot complements.

The main ingredient to understand the topology of the space of smooth long
embeddings is the proof given by A. Hatcher of the Smale Conjecture

######  Theorem 2.5.1 ([Hat83]).

The group of diffeomorphisms $\operatorname{Diff}({\mathbb{D}}^{3})$ of the
$3$–ball fixing the boundary pointwisely is contractible.

One consequence of the previous Theorem is the following

######  Lemma 2.5.2.

There is a homotopy equivalence

(9)
$\Omega{\mathfrak{Emb}}^{\gamma}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})\cong\operatorname{Diff}(C_{\gamma}).$

###### Proof.

The fiber of the fibration
$\operatorname{Diff}({\mathbb{D}}^{3})\rightarrow{\mathfrak{Emb}}^{\gamma}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$
is (homotopy equivalent) to the group $\operatorname{Diff}(C_{\gamma})$. Since
the group $\operatorname{Diff}({\mathbb{D}}^{3})$ is contractible the result
follows. ∎

Thus, to understand the homotopy type of the space
${\mathfrak{Emb}}^{\gamma}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ it is
enough to understand the diffeomorphism group of the manifold $C_{\gamma}$.

####  2.5.2. Embeddings of disks into irreducible $3$-manifolds.

Let $M^{3}$ be a orientable compact connected irreducible $3$–manifold with
boundary. Let $i:{\mathbb{D}}^{2}\hookrightarrow M^{3}$ be a proper embedding
of the $2$–disk into $M$ and consider the space
${\mathfrak{Emb}}({\mathbb{D}}^{2},M^{3})$ of embeddings
$\phi:{\mathbb{D}}^{2}\rightarrow M^{3}$ such that
$\phi_{|{\mathcal{O}p}(\partial{\mathbb{D}}^{2})}=i_{|{\mathcal{O}p}(\partial{\mathbb{D}}^{2})}$.
The following holds

######  Theorem 2.5.3 (Hatcher [Hat99a]).

The space ${\mathfrak{Emb}}({\mathbb{D}}^{2},M^{3})$ is contractible.

######  Corollary 2.5.4.

Let $H^{g}$ be the genus $g$ handlebody. The diffeomorphism group
$\operatorname{Diff}(H^{g})$ is contractible.

###### Proof.

Since $\operatorname{Diff}({\mathbb{D}}^{3})$ is contractible because of
Theorem 2.5.1, it is enough to check that $\operatorname{Diff}(H^{g})$ is
homotopy equivalent to $\operatorname{Diff}(H^{g-1})$ for any $g\geq 1$.
Consider any proper embedding of a separating disk
$e:{\mathbb{D}}^{2}\rightarrow H^{g}$. The postcomposition of $e$ by any
diffeomorphism of $H^{g}$ induces a fibration

$\operatorname{Diff}(H^{g}\backslash
e({\mathbb{D}}^{2}))\rightarrow\operatorname{Diff}(H^{g})\rightarrow{\mathfrak{Emb}}({\mathbb{D}}^{2},H^{g}).$

The fiber can be identified with $\operatorname{Diff}(H^{g-1})$ and the base
is contractible by Theorem 2.5.3 so the result follows. ∎

####  2.5.3. The homotopy type of the space of smooth long embeddings.

######  Theorem 2.5.5 (Hatcher).

The path–connected component
${\mathfrak{Emb}}^{0}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ of the space
of smooth long embeddings containing the unknot is contractible.

###### Proof.

In view of Lemma 2.5.2 it is enough to check that the space
$\operatorname{Diff}({\mathbb{D}}^{2}\times{\mathbb{S}}^{1})$ is contractible.
This follows from Corollary 2.5.4. ∎

In general the following holds

######  Theorem 2.5.6 (Hatcher).

Let $\gamma\in{\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ be
any smooth long embedding. Then, the space
${\mathfrak{Emb}}_{N,jN}^{\gamma}({\mathbb{S}}^{1},{\mathbb{S}}^{3})$ is a
$K(\pi,1)$.

###  2.6. Microfibration Lemma

######  Definition 2.6.1.

A map $p:E\rightarrow B$ is a Serre microfibration if for any $k\geq 0$ and
any pair of maps $h:{\mathbb{D}}^{k}\times[0,1]\rightarrow B$ and
$g:{\mathbb{D}}^{k}\times\\{0\\}\rightarrow E$ such that $p\circ
g=h_{|{\mathbb{D}}^{k}\times\\{0\\}}$ there exists a positive number
$\varepsilon>0$ and a map
$\hat{g}:{\mathbb{D}}^{k}\times[0,\varepsilon]\rightarrow E$ satisfying that

  * •

$\hat{g}_{|{\mathbb{D}}^{k}\times\\{0\\}}=g$ and

  * •

$p\circ\hat{g}=h_{{\mathbb{D}}^{k}\times[0,\varepsilon]}$.

######  Lemma 2.6.2 (Microfibration Lemma [Wei05]).

Let $p:E\rightarrow B$ be a Serre microfibration with weakly contractible
fibers. Then $p$ is a Serre fibration.

##  3\. Eliashberg and Giroux Theorems

We need the parametric versions of two Theorems due to Eliashberg for tight
contact structures in ${\mathbb{S}}^{3}$ or in ${\mathbb{D}}^{3}$ and one due
to Giroux. We claim that the first two are already proven: contractibility of
the space of contact structures in the ball and contractibility of the space
of contactomorphisms [Jän18]222With the tools developed in this article, we
are writing an alternative proof of this result [FFP].. In fact, they are
completely equivalent using Theorem 2.5.1. However, the third and the fourth
statements require a proof. The third one is the heart of the article: a
multiparametric convex surface theory. In fact, we cheat a bit. We use the
contractibility of the space of contactomorphisms of the ball to prove right
away that the space of convex disks with fixed characteristic foliation in the
ball is also contractible. A matter of algebraic topology force game:
diffeomorphisms are contractible in the ball, contactomorphisms are
contractible in the ball, then standard convex disks are contractible in the
ball. It comes in two flavours the already mentioned one and another one for
multiparametric families of standard spheres, where contractibility is not
obtained, since some formal evaluation data needs to be fixed.

###  3.1. Classification of tight contact structures in the $3$–sphere.

We have the following Theorem, due to Eliashberg that completely characterizes
the connectedness of the subspace of tight contact structures homotopic to the
standard one in ${\mathbb{S}}^{3}$.

######  Theorem 3.1.1 (Theorem 2.1.1 in [Eli92]).

A tight contact structure on ${\mathbb{S}}^{3}$ is isotopic to the standard
contact structure $\xi_{{\operatorname{std}}}$

We want to generalize it to the following multiparametric version. Let
${\mathfrak{C}Str}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ be the space of
tight contact structures on ${\mathbb{S}}^{3}$. Fix
$\operatorname{ev}_{N}:{\mathfrak{C}Str}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\to\operatorname{SO}(4)/\operatorname{U}(2)={\mathbb{S}}^{2}$
the evaluation map of the contact structure at the north pole on
$N\in{\mathbb{S}}^{3}$.

######  Theorem 3.1.2 (Theorem 2.1 in [Jän18]).

The map $\operatorname{ev}_{N}$ is a weak homotopy equivalence.

The techniques employed in Eliashberg’s original result- Theorem 3.1.1\- were
believed to imply this multiparametic case. However, this was not written in
the literature. This gap has been filled quite recently in [Jän18]. As a
consequence of the techniques developed in this article we have an alternative
proof of this result, this will be the content of a future article.

###  3.2. Contactomorphisms in the standard contact $3$–sphere.

Consider the evaluation map

(10)
$\operatorname{ev}_{N}:\operatorname{Cont}({\mathbb{S}}^{3},\xi_{{\operatorname{std}}})\to\operatorname{U}(2),$

defined by evaluating the image of the north pole and the jacobian at the
north pole.

Eliashberg again hinted in [Eli92]

######  Corollary 3.2.1.

The evaluation map (10) is a weak homotopy equivalence.

###### Proof.

Just write the following pair of exact sequences, where the vertical lines are
provided by the natural evaluation maps

$\textstyle{\operatorname{Cont}({\mathbb{S}}^{3},\xi_{{\operatorname{std}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diff}({\mathbb{S}}^{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{C}Str}({\mathbb{S}}^{3},\xi_{std})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{U}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{SO}(4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{SO}(4)/\operatorname{U}(2).}$

Use the five Lemma to provide the required answer. ∎

######  Corollary 3.2.2.

The space of compactly supported contactomorphisms of the $3$–ball for the
standard contact structure
$\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})$ is
contractible.

This is an obvious consequence of
$\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})$ being the
fiber of the map (10).

Let
$\gamma\in{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
be any long Legendrian embedding and consider the contact manifold
$(C_{\gamma},\xi_{\operatorname{std}})$.

######  Lemma 3.2.3.

There is a homotopy equivalence

(11)
$\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}})\cong\Omega{\mathfrak{Leg}}^{\gamma}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}}).$

###### Proof.

It follows from Lemma 2.4.5 that the fiber of the fibration
$\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})\rightarrow{\mathfrak{Leg}}^{\gamma}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
is the group $\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}})$. By
Corollary 3.2.2 the group
$\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})$ is
contractible so the result follows. ∎

The following result implies that Theorem 1.0.1 and Theorem 1.0.2 are, in
fact, equivalent.

######  Corollary 3.2.4.

Let
$\gamma\in{\mathfrak{Leg}}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})$
be any long Legendrian embeddings. Then, for any $k\geq 1$, the homomorphism

$\pi_{k}(i):\pi_{k}({\mathfrak{Leg}}^{\gamma}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}}))\rightarrow\pi_{k}({\mathfrak{Emb}}^{\gamma}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3}))$

is injective (surjective) if and only if the homomorphism

$\pi_{k-1}(\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}}))\rightarrow\pi_{k-1}(\operatorname{Diff}(C_{\gamma}))$

induced by the inclusion is injective (surjective).

###### Proof.

The result follows directly from Theorem 2.5.1 and Corollary 3.2.2 by
observing that the following diagram is commutative

$\textstyle{\operatorname{Cont}(C_{\gamma},\xi_{\operatorname{std}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Leg}}^{\gamma}_{N,jN}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diff}(C_{\gamma})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diff}({\mathbb{D}}^{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathfrak{Emb}}^{\gamma}_{N,jN}({\mathbb{S}}^{1},{\mathbb{S}}^{3}).}$

∎

###  3.3. Standard convex disks on a tight contact $3$-fold.

The content of this Subsection is the most important technical tool used in
the proofs of the main results. In particular, we will check that the the
space of convex disks with fixed boundary is homotopy equivalent to the space
of smooth disks with the same boundary condition. The reason to spend time
with the understanding of that space is due to the fact that the homotopy type
of that space completely controls the behaviour of a Giroux $2$-skeleton when
producing the image through a family of contactomorphisms.

Denote by ${\mathbb{B}}^{2}(r)\subseteq{\mathbb{R}}^{2}$ the open ball of
radius $r>0$ with centre at the origin; and by
${\mathbb{D}}^{2}(r)\subseteq{\mathbb{R}}^{2}$ the corresponding closed ball.
When $r=1$ we will simply write ${\mathbb{B}}^{2}$ and ${\mathbb{D}}^{2}$.

Denote by $({\mathbb{D}}^{2},\mathcal{F}_{\operatorname{std}})$ the standard
foliation of the disk with boundary the Legendrian unknot. We will say that an
embedding $e:{\mathbb{D}}^{2}\rightarrow(M^{3},\xi)$ of a disk into a contact
$3$–fold is standard if it is convex and
$({\mathbb{D}}^{2},e^{*}\xi)=({\mathbb{D}}^{2},\mathcal{F}_{\operatorname{std}})$.
In particular, the boundary of $e({\mathbb{D}}^{2})$ is a standard Legendrian
unknot (with $\operatorname{tb}=-1$).

Figure 2. The standard convex disk.

Let $e:{\mathbb{D}}^{2}\rightarrow(M,\xi)$ be an standard embedding of a disk
into a contact $3$–fold. Denote by ${\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$
the space of smooth embeddings $f:{\mathbb{D}}^{2}\rightarrow M$ such that
$f_{|{\mathcal{O}p}(\partial{\mathbb{D}}^{2})}=e_{|{\mathcal{O}p}(\partial{\mathbb{D}}^{2})}$
and $f$ is smoothly isotopic to $e$ (relative to an open neighbourhood of
$\partial{\mathbb{D}}^{2}$). Similarly, define
${\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{D}}^{2},(M,\xi))$ as the
subspace of ${\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$ conformed by embeddings
that are standard.

We will use the following technical definition

######  Definition 3.3.1.

Let $N$ be a compact manifold and $(M,g)$ a Riemannian manifold. Denote by
${\mathfrak{Emb}}^{\mathcal{P}}(N,M)\subseteq{\mathfrak{Emb}}(N,M)$ the
subspace of embeddings that satisfies certain property $\mathcal{P}$. We will
say that the inclusion

$j:{\mathfrak{Emb}}^{\mathcal{P}}(N,M)\hookrightarrow{\mathfrak{Emb}}(N,M)$

is a $C^{0}$-dense homotopy equivalence if

  * •

$j$ is a homotopy equivalence.

  * •

For any $\varepsilon>0$ and any continuous map
$\varphi:K\rightarrow{\mathfrak{Emb}}(N,M)$ from a compact parameter space $K$
there exists an homotopy $\varphi_{t}:K\rightarrow{\mathfrak{Emb}}(N,M)$,
$t\in[0,1]$, such that

    1. (1)

$\varphi_{0}=\varphi$,

    2. (2)

${\operatorname{Image}}(\varphi_{1})\subseteq{\mathfrak{Emb}}^{\mathcal{P}}(N,M)$
and

    3. (3)

$\varphi_{t}(k)(N)\subseteq{\mathbb{B}}(N,\varepsilon)=\\{p\in
M:d_{M}(p,N)<\varepsilon\\}$.

Our main result can be stated as follows.

######  Theorem 3.3.2.

Fix an standard embedding of a disk $e:{\mathbb{D}}^{2}\to(M^{3},\xi)$ into a
tight contact $3$–manifold. The inclusion

${\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{D}}^{2},(M,\xi))\hookrightarrow{\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$

is a $C^{0}$-dense homotopy equivalence.

First, let us state and prove an easy consequence, that will be the main use
of this Theorem. This result solves the case $j=2$ in the argument explained
in the Outline 1.1.

######  Corollary 3.3.3.

Let $e:{\mathbb{D}}^{2}\rightarrow(M,\xi)$ be an standard embedding of a disk
into a tight contact $3$–manifold. Let $\phi_{z}$ a family of embeddings
$\varphi^{z,r}\in\operatorname{Diff}(M;\operatorname{rel}e(\partial{\mathbb{D}}^{2}))$,
$(z,r)\in{\mathbb{D}}^{k}$, be any disk of diffeomorphisms bounding an sphere
of contactomorphisms
$\varphi^{z,1}\in\operatorname{Cont}(M,\xi;\operatorname{rel}e(\partial{\mathbb{D}}^{2}))$,
where $\varphi^{N,1}={\operatorname{Id}}$. Then, there exists a homotopy
$\varphi^{z,r}_{t}\in\operatorname{Diff}(M;\operatorname{rel}e(\partial{\mathbb{D}}^{2}))$,
$t\in[0,1]$, such that

  * (i)

$\varphi^{z,r}_{0}=\varphi^{z,r},$

  * (ii)

$\varphi^{N,1}_{t}={\operatorname{Id}}$,

  * (iii)

$\varphi^{z,1}_{t}\in\operatorname{Cont}(M,\xi;\operatorname{rel}e(\partial{\mathbb{D}}^{2}))$
and

  * (iv)

$\varphi^{z,r}_{1}\in\operatorname{Diff}(M;\operatorname{rel}e({\mathbb{D}}^{2}))$.

What this Corollary states is that if we have an standard disk that is not
preserved by a multiparametric family of contactomorphisms that are isotopic
to the identity as diffeomorphisms, we can deform the family of
contactomorphisms through contactomorphisms in order to make sure that the
deformed family preserves the disk: a technical point is that the new family
is homotopic to the old one. In particular, it is still isotopic to the
identity

###### Proof.

Assume without loss of generality that
$\varphi^{N,r}\equiv{\operatorname{Id}}$, $r\in[0,1]$. This is just to
simplify the notation in the following argument. Denote by $e^{z,r}$ the
postcomposition $e^{z,r}=\varphi^{z,r}\circ e$. Since the inclusion
${\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{D}}^{2},(M,\xi))\hookrightarrow{\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$
is a homotopy equivalence the relative homotopy class

$e^{z,r}\in\pi_{k}({\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M),{\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{D}}^{2},(M,\xi)))$

is trivial. This means that there is an homotopy
$e^{z,r}_{t}\in{\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$, $t\in[0,1]$, such
that

  * •

$e^{z,r}_{0}=e^{z,r}$,

  * •

$e^{z,1}_{t}=e^{z,1}$,

  * •

$e^{N,r}_{t}=e$, $(r,t)\in[0,1]^{2}$,

  * •

$e^{z,r}_{1}=e\in{\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{D}}^{2},(M,\xi))$.

Define the family of vector fields
$X^{z,r}_{t}=\frac{de^{z,1}_{s}}{ds}_{|s=t}$ such that $X^{z,r}_{t}$ is
defined over the disk $e_{z,r,t}$. Use the Isotopy Extension Theorem for
convex surface embeddings to extend the family $X^{z,1}_{t}$ to a family over
the whole manifold $\tilde{X}^{z,1}_{t}$. The relative character of the smooth
extension of isotopy theorem, ensures the existence of a family of smooth
vector fields $\tilde{X}^{z,1}_{t}$ defined over the whole manifold. Integrate
to produce a family of flows $f_{z,r}^{t}$ that are contact for $r=1$. The
family $\varphi_{z,r}^{t}=f_{z,r}^{t}\circ\varphi_{z,r}$ satisfies all the
required properties. ∎

Let us introduce some notation. We say that a map
$\phi:{\mathbb{D}}^{2}\times[-1,1]\to M$ of the closed $3$–ball into a
$3$-fold $M$ is a horned embedding, if

  1. (1)

there is $0<\varepsilon<1/2$ such that restricted to the domain
$U=({\mathbb{D}}^{2}\setminus{\mathbb{D}}^{2}(1-\varepsilon))\times[-1,1]$ the
map can be expressed as $\phi_{|U}=e\circ\pi$, where
$\pi:{\mathbb{D}}^{2}\times[-1,1]\to{\mathbb{D}}^{2}$ is the natural
projection and $e:{\mathbb{D}}^{2}\setminus{\mathbb{D}}^{2}(1-\varepsilon)\to
M$ is an embedding.

  2. (2)

In the complement of $U$ the map $\phi$ is an embedding.

We call $U$ the horned region.

We say that a horned embedding into a contact manifold is semi-standard if the
induced characteristic foliation on ${\mathbb{D}}^{2}\times\\{-1\\}$ is the
standard foliation $({\mathbb{D}}^{2},\mathcal{F}_{{\operatorname{std}}})$.

We say that a horned embedding into a contact manifold is almost-standard if
the induced characteristic foliation on ${\mathbb{D}}^{2}\times\\{-1\\}$ and
on ${\mathbb{D}}^{2}\times\\{1\\}$ are the standard foliation
$({\mathbb{D}}^{2},\mathcal{F}_{{\operatorname{std}}})$.

Finally, we say that the horned embedding is standard if it is almost-standard
and the map is an isocontact embedding once we fix the contact product
structure on the abstract domain ${\mathbb{D}}^{2}\times[-1,1]$

We say that two standard horned embeddings $\phi_{1}$ and $\phi_{2}$ are
almost equivalent if both embeddings do coincide in the boundary of
${\mathbb{D}}^{2}\times[-1,1]$. This is clearly an equivalence relation.

Given a standard horned embedding
$\phi:{\mathbb{D}}^{2}\times[-1,1]\rightarrow(M,\xi)$ we will denote by
$\operatorname{SHE}(\phi)$ the space of standard horned embeddings that are
almost equivalent to $\phi$.

We want to use as a key step in the proof of Theorem 3.3.2 the following
technical result

######  Proposition 3.3.4.

The space $\operatorname{SHE}(\phi)$ is contractible.

###### Proof.

Consider the fibration
$\operatorname{SHE}(\phi)\rightarrow\operatorname{Germs}({\mathbb{D}}^{2}\times\\{-1,1\\}),$
which sends an standard horned embedding $e$ to the germ of embedding over
$\phi({\mathbb{D}}^{2}\times\\{-1,1\\})$. This space can be identified up to
homotopy with the set of contact vector fields transverse to the boundary and
preserving the level sets $e({\mathbb{D}}^{2}\times\\{t\\})$ for
$t\in{\mathcal{O}p}\\{-1,1\\}$. In particular, it is contractible since this
is a convex condition. Thus, it is enough to check that the fiber; i.e. the
space $\operatorname{SHE}(\phi,\partial)$ of almost equivalent standard horned
embeddings that coincide over a germ of ${\mathbb{D}}^{2}\times\\{-1,1\\}$ is
contractible. We will check that it is weakly contractible that is enough for
us. Consider a compact family $e_{k}\in\operatorname{SHE}(\phi,\partial)$,
$k\in K$. Since $K$ is compact, there is a fixed small neighborhood
$\mathcal{N}$ of the boundary $\phi(\partial(D^{2}\times[-1,1]))$ for which
the whole family coincides. Fix an embedded $3$–ball
${\mathbb{B}}^{3}\subseteq\phi({\mathbb{D}}^{2}\times[0,1])\subseteq(M,\xi)$
whose boundary lies in the open neighborhood of the boundary $\mathcal{N}$.
Fix a particular $e_{k_{0}}$ in the family, then,
$\varphi_{k}=e_{k}\circ(e_{k_{0}}^{-1})_{|{\mathbb{B}}^{3}}:({\mathbb{B}}^{3},\xi)\rightarrow({\mathbb{B}}^{3},\xi)$
is a family of contactomorphisms of the tight $3$–ball
$({\mathbb{B}}^{3},\xi)$ that fixes the neighbourhood pointwise. By Corolllary
3.2.2, there exists a family $\varphi_{k,t}$ of contactomorphisms of the ball
such that $\varphi_{k,0}=\varphi_{k}$ and $\varphi_{k,1}={\operatorname{Id}}$.
Since this contactomorphisms are fixed on
${\mathcal{O}p}(\partial{\mathbb{B}}^{3})$ we can extend them to $(M,\xi)$ by
the identity map outside ${\mathbb{B}}^{3}$. Thus, the homotopy

$e_{k,t}=\varphi_{k,t}\circ e_{k_{0}},t\in[0,1],$

is well–defined on the space $\operatorname{SHE}(\phi,\partial)$ and satisfies
that $e_{k,0}=\varphi_{k,0}\circ e_{k_{0}}=\varphi_{k}\circ e_{k_{0}}=e_{k}$
and $e_{k,1}={\operatorname{Id}}\circ e_{k_{0}}=e_{k_{0}}$ is constant. ∎

######  Proposition 3.3.5.

Fix a horned embedding $\phi$ which has boundary
$\phi(\partial{\mathbb{D}}^{2}\times\\{-1\\})$ a Legendrian unknot with
$\operatorname{tb}=-1$. The space of standard disks embeddings inside
$\phi({\mathbb{D}}^{2}\times(-1,1))$ that have fixed boundary
$\phi(\partial{\mathbb{D}}^{2}\times\\{-1\\})$ is contractible.

###### Proof.

Consider any compact family of standard embeddings
$e_{k}:{\mathbb{D}}^{2}\rightarrow\phi({\mathbb{D}}^{2}\times(-1,1))\subseteq(M,\xi)$,
$k\in K$, inside $\phi$. Fix a smooth disk embedding
$f_{0}:{\mathbb{D}}^{2}\rightarrow\phi({\mathbb{D}}^{2}\times(-1,1))$ with
boundary at the Legendrian unknot
$\phi(\partial{\mathbb{D}}^{2}\times\\{-1\\})$ that is disjoint, away from the
boundary, to any standard embedding $e_{k}$. By Giroux Approximation and
Realization Theorems we may assume that $f_{0}$ is standard.

Figure 3. Creating a common convex disk below. Figure 4. Interpolating to the
disk.

Observe that the union of $e_{k}$ and $f_{0}$ conforms an standard horned
sphere embedding inside $\phi$. Moreover, denote by $\mathcal{H}$ the space of
standard horned embeddings inside $\phi$ and by the space $\mathcal{S}$ of
standard horned spheres embedding inside $\phi$. Then, there is a natural
fibration $\mathcal{H}\rightarrow\mathcal{S}$ given by restriction whose fiber
is non–empty (this follows from Schönflies Theorem333Observe that
$\phi({\mathbb{D}}^{2}\times[-1,1])$ is an irreducible $3$–fold and Theorem
3.1.1) and, moreover, contractible by the previous result. Thus, the map
$K\rightarrow\mathcal{S}$ that we have built can be lifted to a map
$K\rightarrow\mathcal{H}$. In other words, there is a family
$g_{k}:{\mathbb{D}}^{2}\times[-1,1]\rightarrow\phi({\mathbb{D}}^{2}\times[-1,1])$
pf standard horned embeddings such that
$(g_{k})_{|{\mathbb{D}}^{2}\times\\{-1\\}}=e_{k}$ and
$(g_{k})_{|{\mathbb{D}}^{2}\times\\{1\\}}=f_{0}$. In particular,
$e_{k,t}=(g_{k})_{|{\mathbb{D}}^{2}\times\\{t\\}}$ defines a homotopy of
standard disks inside $\phi$ between $e_{k,0}=e_{k}$ and $e_{k,1}=f_{0}$.

∎

###### Proof of Theorem 3.3.2.

It is enough to check the following: Let $K$ be any compact parameter space
and $G\subseteq K$ any subspace. Then, for any $\varepsilon>0$ and for any map
$\varphi:K\rightarrow{\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$ such that
$\varphi(k)\in{\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{D}}^{2},(M,\xi))$,
for $k\in G$, there exists an homotopy
$\varphi_{t}:K\rightarrow{\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$,
$t\in[0,1]$, satisfying that

  * •

$\varphi_{0}=\varphi$,

  * •

$\varphi_{t}(k)\in{\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{D}}^{2},(M,\xi))$
for any $(k,t)\in(G\times[0,1])\cup(K\times\\{1\\})$,

  * •

$\varphi_{t}(k)({\mathbb{D}}^{2})\subseteq{\mathbb{B}}(\varphi(k)({\mathbb{D}}^{2}),\varepsilon)$.

To check this proceed as follows. First, consider any continuous extension
$\phi_{k}:{\mathbb{D}}^{2}\times[-1,1]\rightarrow M$, $k\in K$, into a family
of horned embeddings of our initial family $\varphi(k)$, i.e.
$\phi_{k}({\mathbb{D}}^{2}\times\\{0\\})=\varphi(k)({\mathbb{D}}^{2})$, such
that

$\phi_{k}({\mathbb{D}}^{2}\times[-1,1])\subseteq{\mathbb{B}}(\varphi(k)({\mathbb{D}}^{2}),\varepsilon).$

Secondly, we consider the universal space $\mathcal{H}$ of pairs
$(\phi,P_{s})$, $s\in[0,1]$, where
$\phi:{\mathbb{D}}^{2}\times[-1,1]\rightarrow M$ is a horned embedding with
$\phi_{|{\mathcal{O}p}(\partial{\mathbb{D}}^{2}\times\\{0\\})}=e_{|{\mathcal{O}p}(\partial{\mathbb{D}}^{2})}$
and

$P_{s}\in{\mathfrak{Emb}}^{\phi_{|{\mathbb{D}}^{2}\times\\{0\\}}}({\mathbb{D}}^{2},\phi({\mathbb{D}}^{2}\times[-1,1]))$

is an isotopy joining $P_{0}=\phi_{|{\mathbb{D}}^{2}\times\\{0\\}}$ with an
standard disk
$P_{1}:{\mathbb{D}}^{2}\rightarrow\phi({\mathbb{D}}^{2}\times[-1,1])\subseteq(M,\xi)$.
The forgetful map $\mathcal{H}\rightarrow\mathcal{HE},(\phi,P_{s})\mapsto\phi$
over the space of horned embeddings is a microfibration with non–empty
contractible fiber. The contractibility of the fiber follows from Theorem
2.5.3 and Proposition 3.3.5. It follows from the Microfibration Lemma 2.6.2
that then $\mathcal{H}\rightarrow\mathcal{HE}$ is, in fact, a fibration and,
thus, a weak homotopy equivalence.

We have a well–defined map $\phi:K\rightarrow\mathcal{HE},k\mapsto\phi_{k}$
that admits a section over $G\subseteq K$: the constant one given by

$G\rightarrow\mathcal{H},k\mapsto(\phi_{k},P^{k}_{s}\equiv(\phi_{k})_{|{\mathbb{D}}^{2}\times\\{0\\}}).$

It follows that we can extend this section over the whole parameter space $K$
to obtain

$\hat{\phi}:K\rightarrow\mathcal{H},k\mapsto(\phi_{k},P^{k}_{s}).$

The desired homotopy
$\varphi_{t}:K\rightarrow{\mathfrak{Emb}}^{e}({\mathbb{D}}^{2},M)$,
$t\in[0,1]$, is defined as

$\varphi_{t}(k)=P^{k}_{t}.$

∎

###  3.4. The space of standard convex spheres in a tight $3$-fold.

In this Subsection, we will recreate the arguments of Subsection 3.3, for
another case. We will provide the details of the standard convex sphere now.

The careful reader will be able to outline the general method that one can try
to implement to cover most of the cases, from these two examples.

Let ${\mathfrak{Emb}}_{\operatorname{std}}({\mathbb{S}}^{2},(M^{3},\xi))$ be
the space of convex embeddings $e:{\mathbb{S}}^{2}\rightarrow(M,\xi)$ that are
standard, this just means that the characteristic foliation $e^{*}\xi$
coincides with the induced at the boundary of a Darboux ball. Given an
standard embedding $e:{\mathbb{S}}^{2}\rightarrow(M,\xi)$ we will denote by
${\mathfrak{Emb}}^{e}({\mathbb{S}}^{2},M)$ the space of embeddings smoothly
isotopic to $e$ and by
${\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{S}}^{2},(M,\xi))$ the
subspace of ${\mathfrak{Emb}}^{e}({\mathbb{S}}^{2},M)$ conformed by standard
convex embeddings. The following can be thought as a multiparametric version
of Colin’s result in [Col97].

######  Theorem 3.4.1.

Fix any convex standard embedding of the sphere
$e:{\mathbb{S}}^{2}\to(M^{3},\xi)$ as base point. Then,

$\pi_{k}({\mathfrak{Emb}}^{e}({\mathbb{S}}^{2},M),{\mathfrak{Emb}}^{e}_{\operatorname{std}}({\mathbb{S}}^{2},(M,\xi)))=\pi_{k}(\operatorname{SO}(3),\operatorname{U}(1)),$

for any $k\geq 1$.

######  Remark 3.4.2.

It is worth it to mention that if $M$ is an irreducible $3$-fold the statement
can be easily deduced from Theorems 2.5.1 and Corollary 3.2.2. Here is the
argument:

  1. (1)

The fibration

$\operatorname{Diff}({\mathbb{D}}^{3})\rightarrow{\mathfrak{Emb}}({\mathbb{D}}^{3},M)\rightarrow{\mathfrak{Emb}}({\mathbb{S}}^{2},M)$

has contractible fiber because of Theorem 2.5.1.

  2. (2)

The same holds in the contact category since the fiber of the fibration

$\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}})\rightarrow{\mathfrak{CEmb}}(({\mathbb{D}}^{3},\xi_{\operatorname{std}}),(M,\xi))\rightarrow{\mathfrak{Emb}}_{\operatorname{std}}({\mathbb{S}}^{2},(M,\xi)$

is also contractible because of Corollary 3.2.2.

  3. (3)

The space ${\mathfrak{Emb}}({\mathbb{D}}^{3},M)$ is homotopy equivalent to the
oriented frame bundle $\operatorname{Fr}^{*}(M)$ and the space of Darboux
balls ${\mathfrak{CEmb}}(({\mathbb{D}}^{3},\xi_{\operatorname{std}}),(M,\xi))$
to the space of contact framings $\operatorname{CFr}(M,\xi)$. See Subsection
2.3.

  4. (4)

Finally,

$\pi_{k}({\mathfrak{Emb}}({\mathbb{S}}^{2},M),{\mathfrak{Emb}}_{\operatorname{std}}({\mathbb{S}}^{2},(M,\xi)))\cong\pi_{k}(\operatorname{Fr}^{+}(M),\operatorname{CFr}(M,\xi))\cong\pi_{k}(\operatorname{SO}(3),\operatorname{U}(1))$

for any $k\geq 1$.

The proof in the general case is almost a word by word adaptation of Theorem
3.3.2, that is the version for standard convex disks. As in the case of disks,
a similar $C^{0}$–closedness statement holds for relative classes that are
trivial. It also possible to deduce the same statements for any convex sphere
in a tight contact $3$–fold.

###### Proof.

Fix $k\geq 1$. Consider any element
$e^{z,r}\in\pi_{k}({\mathfrak{Emb}}({\mathbb{S}}^{2},M),{\mathfrak{Emb}}^{\operatorname{std}}({\mathbb{S}}^{2},(M,\xi)))$.
Consider the map $\operatorname{ev}_{N}:{\mathbb{D}}^{k}\rightarrow
M,(z,r)\mapsto e^{z.r}(N)$. Then, the pullback bundles
$\operatorname{ev}_{N}^{*}\xi$ and $\operatorname{ev}_{N}^{*}TM$ over the disk
${\mathbb{D}}^{k}$ are trivializable in a unique way, up to homotopy. Identify
$\operatorname{ev}_{N}^{*}\xi={\mathbb{D}}^{k}\times{\mathbb{C}}\subseteq\operatorname{ev}_{N}^{*}TM={\mathbb{D}}^{k}\times{\mathbb{R}}^{3}$.
Thus, the evaluation of the $1$–jet of the family of embeddings provides a
morphism

$D\operatorname{ev}_{N}:{\mathbb{D}}^{k}\rightarrow\operatorname{SO}(3),(z,r)\mapsto
d_{N}e^{z,r}$

which takes values at $\operatorname{U}(1)$ over $\partial{\mathbb{D}}^{k}$.
I.e. it defines a class in
$\pi_{k}(\operatorname{SO}(3),\operatorname{U}(1))$. Define

$\Phi_{k}(e^{z,r})=[D\operatorname{ev}_{N}]\in\pi_{k}(\operatorname{SO}(3),\operatorname{U}(1)),$

it is straightforward to check that this assignement defines a group morphism

$\Phi_{k}:\pi_{k}({\mathfrak{Emb}}({\mathbb{S}}^{2},M),{\mathfrak{Emb}}^{\operatorname{std}}({\mathbb{S}}^{2},(M,\xi)))\rightarrow\pi_{k}(\operatorname{SO}(3),\operatorname{U}(1)).$

The homomorphism $\Phi_{k}$ is surjective. Indeed, let
$A^{z,r}\in\pi_{k}(\operatorname{SO}(3),\operatorname{U}(1))$ be any element
of the relative homotopy group. Then, $e^{z,r}=e\circ A^{z,r}$ satisfies that
$\Phi_{k}(e^{z,r})=A^{z,r}$.

It remains to check that $\Phi_{k}$ is injective. Let
$e^{z,r}\in\ker(\Phi_{k})$. Assume without loss of generality that
$e^{N,r}=e$. It is easy to arrange, after a possible contact isotopy, that

$e^{z,r}(N)\equiv e(N)$

is constant and, since $e^{z,r}\in\ker(\Phi_{k})$, that

$d_{N}\tilde{e}^{z,r}\equiv d_{N}e$

is also constant.

Extend the whole family $e^{z,r}$ into a family of spherical annulus
embeddings

$\phi^{z,r}:{\mathbb{S}}^{2}\times[-1,1]\rightarrow(M,\xi),$

such that, $\phi^{z,r}_{|{\mathbb{S}}^{2}\times\\{0\\}}=e^{z,r}$ by using the
normal vector field of each sphere. Observe that by hypothesis the extension
near the north pole is done by using the same vector field which is actually a
contact vector field. In particular, the evaluation of the differential map of
each sphere $\phi^{z,r}_{t}=\phi^{z,r}_{|{\mathbb{S}}^{2}\times\\{t\\}}$ is
constant by construction.

Now follow word by word the setup explained for disks. Define in the same way
an almost standard and standard spherical annulus embedding
$\phi:{\mathbb{S}}^{2}\times[-1,1]\rightarrow(M,\xi)$. Also define the
equivalence relation between standard annulus embeddings of being almost
equivalent.

The technical assumption that is going to be implicit in the following Lemmas
is that the evaluation of the differential of each sphere $\phi_{t}$ is
constant.

The following Lemma replaces Proposition 3.3.4 in the disk case.

######  Lemma 3.4.3.

The space of standard spherical annulus embeddings almost equivalent to a
fixed one $\phi$ is contractible.

###### Proof.

The proof is similar to the proof of Proposition 3.3.4. This space can be
identified with the space of contactomorphisms of
${\mathbb{S}}^{2}\times[-1,1]$ that have trivial evaluation over
$\\{N\\}\times[-1,1]$. It is a consequence of Corollary 3.2.2 that this group
is contractible. ∎

In the same way, the following result plays the role of Proposition 3.3.5.

######  Lemma 3.4.4.

Let $\phi:{\mathbb{S}}^{2}\times[-1,1]\rightarrow(M,\xi)$ be any spherical
annulus. The space of standard spheres inside
$\phi({\mathbb{S}}^{2}\times[-1,1])$ smoothly isotopic to
$\phi({\mathbb{S}}^{2}\times\\{0\\})$ and with trivial evaluation at the north
pole is contractible.

###### Proof.

The proof is analogous to that of Proposition 3.3.5. ∎

As in the proof of Theorem 3.3.2 define a microfibration
$\mathcal{S}\rightarrow\mathcal{SA}$, where $\mathcal{S}$ is the space of
pairs $(\phi,P_{s})$ of spherical annulus embeddings
$\phi:{\mathbb{S}}^{2}\times[-1,1]\rightarrow(M,\xi)$ with trivial evaluation
at the north pole and
$P_{s}:{\mathbb{S}}^{2}\rightarrow\phi({\mathbb{S}}^{2}\times[-1,1])$,
$s\in[0,1]$, is a smooth isotopy of spheres with trivial evaluation at the
north pole joining $P_{0}=\phi_{0}$ and a standard sphere $P_{1}$. The fiber
of this microfibration is contractible by the previous Lemma444It is a
consequence of the Smale conjecture that the space of smooth embeddings
${\mathbb{S}}^{2}\rightarrow{\mathbb{S}}^{2}\times[-1,1]$ with trivial
evaluation at the north pole and smoothly isotopic to the inclusion
${\mathbb{S}}^{2}\hookrightarrow{\mathbb{S}}^{2}\times\\{0\\}\subseteq{\mathbb{S}}^{2}\times[-1,1]$
is contractible.. By the Microfibration Lemma 2.6.2 the map
$\mathcal{S}\rightarrow\mathcal{SA}$ is a fibration with contractible fiber
and, thus, a homotopy equivalence.

The initial family of spherical annulus embeddings

$\phi^{z,r}\in\mathcal{SA}$

admits a lift to $\mathcal{S}$ over the boundary $\\{r=1\\}$, namely the
constant one

$(\phi^{z,1},P^{z,1}_{s}\equiv\phi^{z,1}_{0}).$

Fix any extension of this lift to the whole disk $(\phi^{z,r},P^{z,r}_{s})$.
The family

$e^{z,r}_{t}=P^{z,r}_{t},t\in[0,1],$

defines an homotopy between the initial disk $e^{z,r}$ and a disk of standard
spheres $P^{z,r}_{1}$. This concludes the proof of the injectivity of
$\Phi_{k}$ and, thus, of the result. ∎

######  Remark 3.4.5.

The general idea behind the proof of Theorems 3.3.2 and 3.4.1 is that whenever
we have control over the homotopy type of
$\operatorname{Diff}(\Sigma\times[-1,1])$ and
$\operatorname{Cont}(\Sigma\times[-1,1],\xi)$ for any convex surface
$e:\Sigma\rightarrow(M,\xi)$ inside a tight $3$-fold, i.e. the symmetries of
the tubular neighbourhoods of the embeddings, we can recreate the previous
arguments to obtain similar statements. A technical point is that one should
be able to ensure the existence of at least one convex embedding contact
isotopic to $\Sigma$ in each tubular neighbourhood of the form
$\phi(\Sigma\times[-1,1])$. This follows directly from Giroux Theory in the
case of the standard disk and sphere but it is not clear in general.

##  4\. Reduction relative to a Giroux-handlebody.

###  4.1. Contact cell decompositions.

The suitable cell decomposition of a tight contact $3$–manifold $(M^{3},\xi)$
that we are going to use was introduced by E. Giroux [Gir02].

######  Definition 4.1.1 (Giroux).

Let $(M^{3},\xi)$ be a compact contact $3$–manifold. A contact cell
decomposition of $(M^{3},\xi)$ is a cell decomposition satisfying the
following properties

  1. (i)

The $1$–skeleton is a Legendrian graph

  2. (ii)

Each $3$–cell is a Darboux ball and

  3. (iii)

Each $2$–cell is attached to a Legendrian unknot with $\operatorname{tb}=-1$.

Let $(M^{3},\xi)$ be any compact contact $3$-manifold equipped with a contact
cell decomposition. We will denote the $i$–skeleton of $(M^{3},\xi)$ by

$G_{i}(M^{3},\xi)\subseteq(M^{3},\xi).$

We will also write $G_{i}$ instead of $G_{i}(M^{3},\xi)$ whenever is clear.

It turns out that $G_{1}$ is just an embedded Legendrian graph inside
$(M^{3},\xi)$. Thus, an small regular neighbourhood of $G_{1}$ is an embedded
handlebody in $M$. We will say that ${\mathcal{O}p}(G_{1})$ is a Giroux
handlebody of $(M,\xi)$. The main result in this Section is that, in some
sense, all the interesting Contact Topology of a tight contact $3$-fold
$(M,\xi)$ is captured by $G_{1}$.

It is well known that any compact contact $3$–manifold admits a contact cell
decomposition. It is also possible to construct the decomposition in such a
way that it contains a prescribed Legendrian link in the $1$–skeleton. In this
case we say that the contact cell decomposition is adapted to the Legendrian
link. We are going to use this fact only in
$({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ so we will provide a proof:

######  Lemma 4.1.2.

Let $L\subseteq({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ be any embedded
connected Legendrian. Then, there exists a contact cell decomposition of
$({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ adapted to $L$. Moreover, the
contact cell decomposition can be constructed in such a way that
$G_{0}({\mathbb{S}}^{3},\xi_{\operatorname{std}})\subseteq L$.

###### Proof.

Since $({\mathbb{S}}^{3}\backslash\\{p\\},\xi_{\operatorname{std}})$ is
contactomorphic to $({\mathbb{R}}^{3},\xi_{\operatorname{std}})$ we may assume
that $L$ lies in the standard contact
$({\mathbb{R}}^{3},\xi_{\operatorname{std}})$. Thus, we have a well–defined
front projection of $L$ into ${\mathbb{R}}^{2}$. Recall that the front
projection is a Legendrian fibration. Add the double points of the projection
to the $0$–skeleton and each Legendrian segment joining them in the fiber
direction to the $1$–skeleton. It follows that the $1$–skeleton that we have
defined is the core of a handlebody in ${\mathbb{S}}^{3}$, i.e. a bunch of
Legendrian unknots. If all these Legendrian unknots have
$\operatorname{tb}=-1$ then we are done. If this is not the case there must be
some Legendrian unknot $U$ in the $1$–skeleton that has
$\operatorname{tb}=-k<-1$. It follows from Eliashberg-Fraser Theorem [EF09]
that we can find $k-1$ stabilizations in $U$. Pick one stabilization
$S\subseteq U$. We may assume that the front of $S$ has the usual zig-zag
picture with $2$ cusps $c_{1}$ and $c_{2}$. Each of these cusps has two
well–defined angles $\alpha_{0}^{i}$ and $\alpha_{1}^{i}$, $i\in\\{1,2\\}$,
such that $\alpha_{0}^{i}+\alpha_{1}^{i}=2\pi$. Assume that
$\alpha_{0}^{i}<\pi<\alpha_{2}^{i}$. The front of $U$ divides
${\mathbb{R}}^{2}$ in two regions one bounded $\mathcal{B}$ and one unbounded
$\mathcal{U}$. There is one, and only one, cusp $c_{i}$, $i\in\\{0,1\\}$, of
the two such that the angle of the boundary of $B$ around $c_{i}$ is
$\alpha_{1}^{i}$ and the other cusp satisfies the opposite. We can always find
another cusp $c_{j}$ in $U$ such that there is a curve $a_{ij}$ connecting
$c_{i}$ to $c_{j}$ contained in $B$. If $c_{i}$ (respectively $c_{j}$) is a
cusp of the original knot, we place an extra vertex on it. If not, there are
two vertices projecting over the same cusp and we choose any of the two.
Since, we need to lift $a_{ij}$ to a Legendrian curve, that will be a new
$1$-cell in the skeleton and this why we need to choose one of the two
vertices. This $1$–cell breaks $U$ into two Legendrian unknots $U_{1}$ and
$U_{2}$ such that both of them have less than $k-1$ stabilizations. Repeat the
process inductively to kill all the stabilizations. Thus, after possibly
adding more $0$–cells and $1$–cells, we may assume that the $1$–skeleton is
the core of a handlebody in ${\mathbb{S}}^{3}$ constructed from a bunch of
Legendrian unknots with $\operatorname{tb}=-1$. By construction all the
$0$–cells are contained in $L$.

Each bounded connected component of the front projection is a disk that lifts
to an embedded disk that bounds a Legendrian unknot with
$\operatorname{tb}=-1$.

The remaining region is a $3$-ball that by Theorem 6.10.2 is Darboux. ∎

(a) Adding vertical Legendrians arcs to reduce to unknots

(b) Isolating the unknots.

Figure 5. First steps of the algorithm explained in Lemma 4.1.2

(a) Model 1 of cancellation of a stabilization

(b) Model 2 of cancelation of a stabilization .

Figure 6. Last steps of the algorithm explained in Lemma 4.1.2: reducing to
$tb=-1$ unknots

###  4.2. Main reduction Theorem.

######  Theorem 4.2.1.

Let $(M^{3},\xi)$ be a compact tight contact $3$-manifold equipped with a
contact cell decomposition. Then, the inclusion
$\operatorname{Cont}(M^{3},\xi;\operatorname{rel}G_{1})\hookrightarrow\operatorname{Diff}(M^{3};\operatorname{rel}G_{1})$
is a homotopy equivalence. In particular, the group
$\operatorname{Cont}(M,\xi;\operatorname{rel}G_{1})$ is contractible.

To derive Theorem 1.0.4 from this result we must be able to check that any
sphere of contactomorphisms that is smoothly contractible can be homotoped to
lie in the subgroup of contactomorphisms that fixes the Giroux handlebody.
Observe that this in particular implies that the formal structure is only used
in the $1$–skeleton.

###### Proof.

Let $\varphi^{z}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{1})$,
$z\in{\mathbb{S}}^{k}$, be an sphere of contactomorphisms that is contractible
inside $\operatorname{Diff}(M;\operatorname{rel}G_{1})$, i.e. there exists a
disk $\varphi^{z,r}\in\operatorname{Diff}(M;\operatorname{rel}G_{1})$,
$r\in[0,1]$, such that $\varphi^{z,1}=\varphi^{z}$ and
$\varphi^{N,r}={\operatorname{Id}}$. Consider a $2$–cell of the contact cell
decomposition
$e:({\mathbb{D}}^{2},\partial{\mathbb{D}}^{2})\rightarrow(M,G_{1})$. Apply
Corollary 3.3.3 to find an homotopy
$\varphi^{z,r}_{t}\in\operatorname{Diff}(M;\operatorname{rel}G_{1})$,
$t\in[0,1]$, such that

  * •

$\varphi^{z,r}_{0}=\varphi^{z,r}$,

  * •

$\varphi^{z,1}_{t}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{1})$,

  * •

$\varphi^{N,r}_{t}={\operatorname{Id}}$

  * •

$\varphi^{z,r}_{1}\in\operatorname{Diff}(M;\operatorname{rel}G_{1}\cup{\mathcal{O}p}(e)).$

Proceed inductively over the rest of $2$–cells to end up with a family
$\widehat{\varphi}^{z,r}\in\operatorname{Diff}(M;\operatorname{rel}G_{2})$
such that
$\widehat{\varphi}^{z,1}\in\operatorname{Cont}(M;\operatorname{rel}G_{2})$.
Observe that the group $\operatorname{Cont}(M;\operatorname{rel}G_{2})$ can be
identified with the disjoint union of $N$ copies of
$\operatorname{Cont}({\mathbb{D}}^{3},\xi_{\operatorname{std}};\operatorname{rel}\partial)$
so the result now follows from Corollary 3.2.2. The contractibility follows
from Corollary 2.5.4. ∎

Let us extract an useful consequence that will be used several times in the
next pages. It is a parametric version of the main result of [EF09], it reads
as follows.

######  Corollary 4.2.2.

The path-connected component of the space of long Legendrian embeddings
containing the unknot with $\operatorname{tb}=-1$ is contractible.

###### Proof.

It is completely equivalent, according to Corollary 3.2.4, to show that the
space of contactomorphims that preserve a neighborhood of the unknot in
${\mathbb{S}}^{3}$ homotopically injects in the space of diffeomorphisms that
preserve the same neighborhood. Indeed, the space of diffeomorphism of the
solid torus that preserves pointwisely a neighbourhood of the boundary is
contractible (Lemma 2.5.5).

But, this is deduced from the fact that there is a contact cell decomposition
of $({\mathbb{S}}^{3},\xi_{\operatorname{std}})$ where the $1$-skeleton is
just the standard Legendrian unknot with $\operatorname{tb}=-1$.We just need
to assume that there are $4$ cells: one $0$–cell that will be a point of the
unknot, a $1$-cell that will be the rest of the unknot, a $2$-cell a Seifert
disk of the unknot and a $3$-cell that is that the rest of the sphere.
Obviously if we assume that our space of contactomorphims/diffeomorphisms
preserves a neighborhood of a Legendrian unknot, it satisfies the hypothesis
of Theorem 4.2.1. ∎

######  Remark 4.2.3.

This result looks surprising. It is obvious that the space of formal long
Legendrian unknots is not contractible, it has a very non-trivial topology.
What the previous result shows is that there is a canonical choice of formal
structure that can be understood as a canonical global section of the
fibration

(12)
${\mathfrak{FLeg}}_{N,jN}({\mathbb{S}}^{3},\xi_{{\operatorname{std}}})\to{\mathfrak{Emb}}_{N,jN}({\mathbb{S}}^{3}).$

In other words, there is a canonical choice of formal Legendrian structure for
any sphere of long smooth embeddings: i.e. this can be understoood like a very
rigid generalization of the Thurston-Benequinn inequality for higher order
homotopy groups. In fact, we will prove a generalization of Corollary 4.2.2 in
Theorem 1.0.3 that states also that there is a section between long smooth
embeddings and long formal Legendrian embeddings: showing that the picture is
completely rigid for any kind of smooth long embeddings.

It follows from Eliashberg-Fraser Theorem that any other Legendrian unknot is
obtained by a sequence of stabilizations from the standard Legendrian unknot
with $\operatorname{tb}=-1$. In this cases, the contact cell decomposition has
more $1$–cells than just the Legendrian unknot so the previous result does not
apply directly.

As a consequence of the previous result we can generalize a Theorem of Giroux
[Gir01b] about the space of contact elements over the disk. We need the
following

######  Lemma 4.2.4 ([DG07]).

The complement of a standard Legendrian unknot $L\subseteq{\mathbb{S}}^{3}$ is
contactomorphic to the space of contact elements over the disk.

The statement follows by observing that the $3$–sphere admits a Legendrian
Hopf fibration ${\mathbb{S}}^{3}\rightarrow{\mathbb{S}}^{2}$ where the fibers
are Legendrian unknots with $\operatorname{tb}=-1$. Thus, the complement of
one of the fibers is a Legendrian fibration over the $2$-disk.

######  Corollary 4.2.5.

The contactomorphism group of the space of contact elements over the disk is
contractible.

###  4.3. Deformation at the $0$–skeleton.

######  Lemma 4.3.1.

Let $(M,\xi)$ be any compact contact $3$–manifold equipped with a contact cell
decomposition and fix $p\in G_{0}$ in the $0$–skeleton. Let $K$ be a compact
parameter space and $B\subseteq K$ a subspace. Let
$\varphi^{k}\in\operatorname{Diff}(M,\xi;\operatorname{rel}p)$, $k\in K$, be
any family of diffeomorphisms such that
$\varphi^{k}\in\operatorname{Cont}(M,\xi;\operatorname{rel}p)$, for $k\in B$.
Then, there exists a homotopy
$\varphi^{k,t}\in\operatorname{Diff}(M,\xi;\operatorname{rel}p)$, $t\in[0,1]$,
satisfying that

  * •

$\varphi^{k,0}=\varphi^{k}$,

  * •

$\varphi^{k,t}\in\operatorname{Cont}(M,\xi;\operatorname{rel}p)$ for any
$(k,t)\in B\times[0,1]$,

  * •

$\varphi^{k,1}\in\operatorname{Diff}(M,\xi;\operatorname{rel}G_{0})$

###### Proof.

Denote the points in the $0$–skeleton by $G_{0}=\\{p,p_{1},\ldots,p_{N}\\}$.
Fix a family of points
$\\{p^{\prime}_{1},\ldots,p^{\prime}_{N}\\}\in{\mathcal{O}p}(p)$ such that
$\varphi^{k}_{|{\mathcal{O}p}(p^{\prime}_{j})}={\operatorname{Id}}$, for any
$j\in\\{1,\ldots,N\\}$, and a contact isotopy
$f_{t}\in\operatorname{Cont}(M,\xi;\operatorname{rel}p)$, $t\in[0,1]$, such
that $f_{0}={\operatorname{Id}}$ and $f_{1}(p^{\prime}_{j})=p_{j}$ for any
$j\in\\{1,\ldots,N\\}$.

The family

$\varphi^{k,t}=f_{t}\circ\varphi^{k}\circ f_{t}^{-1},t\in[0,1],$

provides the required homotopy. ∎

###  4.4. Deformation at the $1$–skeleton.

In order to derive Theorems 1.0.1 and 1.0.4 from Theorem 4.2.1 we must be able
to prove Step $4$ in Subsection 1.1 in the case $j=1$; i.e.

$\pi_{k}(\operatorname{Cont}(M,\xi,\operatorname{rel}G_{0}))$

is mapped injectively into

$\pi_{k}(\operatorname{Diff}(M;\operatorname{rel}G_{0}))$

and any element

$\beta\in\pi_{k}(\operatorname{Diff}(M;\operatorname{rel}G_{0}))$

that admits a formal contact structure is homotopic to an element in

$\pi_{k}(\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0})).$

Assuming the result for $j=1$.

The content of the following two results solves this step for $k\geq 1$.

######  Proposition 4.4.1.

Let $(M,\xi)$ be any tight compact contact $3$–manifold equipped with a
contact cell decomposition. Let
$\varphi^{z}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0})$,
$z\in{\mathbb{S}}^{k}$, $k>0$, be any sphere of contactomorphisms that is
contractible inside $\operatorname{Diff}(M,\xi;\operatorname{rel}G_{0})$.
Then, there exists a homotopy
$\varphi^{z}_{t}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0})$,
$t\in[0,1]$, starting at $\varphi^{z}=\varphi^{z}_{0}$ and such that
$\varphi^{z}_{1}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{1})$.

###### Proof.

We will provide a proof in the case of a unique $1$–cell $l$ in the
$1$–skeleton $G_{1}$. The argument readily applies to the case of a finite
number of $1$–cells without any changes.

Consider a disk of diffeomorphisms
$\varphi^{z,r}\in\operatorname{Diff}(M;\operatorname{rel}G_{0})$ bounding the
sphere $\varphi^{z}$. Assume for notational simplicity that
$\varphi^{N,r}={\operatorname{Id}}$, $r\in[0,1]$.

Denote the postcomposition of the $1$-cell $l$ by the family $\varphi^{z}$ by

$l^{z}=\varphi^{z}\circ l.$

Observe that $l^{z}$ is a family of Legendrians such that
$l^{z}_{|{\mathcal{O}p}\\{-1,1\\}}=l_{|{\mathcal{O}p}\\{-1,1\\}}$ which is
smoothly contractible by hypothesis. This means that there exists a disk

$l^{z,r}=\varphi^{z,r}\circ l$

of formal Legendrians, bounding $l^{z}$, which are actually Legendrian on
$l^{z,r}_{|{\mathcal{O}p}\\{-1,1\\}}$.

Fix $\varepsilon>0$ small enough such that
$l^{z,r}_{|[-1,-1+2\varepsilon]\cup[1-2\varepsilon,1]}$ is constant. Consider
also a small Weinstein chart
$e:({\mathbb{B}}^{3},\xi_{\operatorname{std}})\rightarrow(M,\xi)$ around $l$
such that

$e(x,0,0)=l(x).$

Define the family of smooth charts $e^{z,r}=\varphi^{z,r}\circ e$. Now fix the
family of smooth embeddings of
$s^{z,r}=e_{|\partial{\mathbb{B}}^{3}(1-\varepsilon)}^{z,r}:{\mathbb{S}}^{2}\rightarrow
M$. Further denote the base point by $s=s^{N,r}$, $r\in[0,1]$. Observe that
$s^{z,r}$ defines an element in
$\pi_{k+1}({\mathfrak{Emb}}^{s}({\mathbb{S}}^{2},M),{\mathfrak{Emb}}^{s}_{\operatorname{std}}({\mathbb{S}}^{2},(M,\xi)))$.

Apply Theorem 3.4.1 to obtain a homotopic $C^{0}$-close family of standard
spheres $\tilde{s}^{z,r}$ such that $s^{z,1}=\tilde{s}^{z,1}$. Apply the
Isotopy Extension Theorem for surfaces with fixed characteristic foliation to
find a family of contactomorphisms
$g^{z,r}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0})$ such that

  * •

$g^{N,r}={\operatorname{Id}}$,

  * •

$g^{z,r}\circ\tilde{s}^{z,r}=s$.

Define

$\varphi^{z}_{t}=g^{z,t}\circ\varphi^{z}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0}),t\in[0,1].$

Observe that by construction $\varphi^{z}_{1}\circ l[-1,1]\subseteq
e({\mathbb{B}}^{3})$. In other words, $\varphi^{z}_{1}\circ l$,
$z\in{\mathbb{S}}^{k}$, can be regarded as a family of long standard
Legendrian unknots in a Darboux ball and, thus, by Corollary 4.2.2, it is
contractible. It is important to recall that here we are using that $k>0$. In
the case $k=0$ we must argue that both unknots are Legendrian isotopic which
is not the case in general.

Thus, there exists another family of contactomorphisms
$f^{z,r}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0})$, such that
$f^{N,r}={\operatorname{Id}}$ and $f^{z,1}\circ\varphi^{z}_{1}\circ l=l$.
Define the new homotopy

$\widetilde{\varphi}^{z}_{t}=f^{z,t}\circ\varphi^{z}_{1}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0}),t\in[0,1].$

It follows that
$\widetilde{\varphi}^{z}_{1}\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{1})$
and the problem is solved. ∎

######  Remark 4.4.2.

  * •

Do note that the family $\varphi^{z,r}$, $r<1$ is not pushed through the
homotopy in the previous proof. The reason is that if we try to do it, we can
eventually do the whole process. However, it may be that the family $g^{z,t}$
when composed with $\varphi^{z,r}$ is not the identity on a tubular
neighborhood of the $1$-cell. This is true for $r=1$ since the composition map
$g^{z,1}\circ\varphi^{z,1}$ is a contactomorphism and so, it preserves the
contact framing of $l$. So, we really need a contactomorphism or at least a
formal contactomorphism in order to do that.

  * •

The case $k=0$ is solved if we assume that the contactomorphisms
$\varphi\in\operatorname{Cont}(M,\xi;\operatorname{rel}G_{0})$ is formally
isotopic to the identity; i.e. inside
$\operatorname{FCont}(M,\xi;\operatorname{rel}G_{0})$. Indeed, in this case
the initial $1$–cell $l$ and $\varphi(l)$ have the same formal invariants so,
once we have deformed $\varphi(l)$ to lie in the same Darboux chart than $l$,
Eliashberg-Fraser Theorem does the job [EF09].

######  Proposition 4.4.3.

Let $(M,\xi)$ be any tight compact contact $3$–manifold equipped with a
contact cell decomposition. Let $\varphi^{z}\in D_{k}$,
$z\in{\mathbb{S}}^{k}$, be any sphere of diffeomorphisms that admits a formal
contact structure. Then, there exists a homotopy
$\varphi^{z}_{t}\in\operatorname{Diff}(M;\operatorname{rel}G_{0})$,
$t\in[0,1]$, starting at $\varphi^{z}=\varphi^{z}_{0}$ and such that
$\varphi^{z}_{1}\in\operatorname{Diff}(M;\operatorname{rel}G_{0})$ is a
contactomorphism over ${\mathcal{O}p}(G_{1})$.

###### Proof.

The proof follows closely the argument for the injective case: Proposition
4.4.1. Deal just with one $1$-cell like in the other case.

Consider a sphere of diffeomorphisms
$\varphi^{z}\in\operatorname{Diff}(M;\operatorname{rel}G_{0})$ that, by
hypothesis, can be equipped with a formal contact structure. Assume that
$\varphi^{N}={\operatorname{Id}}$.

Denote the postcomposition of the $1$-cell $l$ by the family $\varphi^{z}$ by

$l^{z}=\varphi^{z}\circ l$

Observe that $l^{z}$ is a family of formal Legendrians such that
$l^{z}_{|{\mathcal{O}p}\\{-1,1\\}}=l_{|{\mathcal{O}p}\\{-1,1\\}}$ .

Fix $\varepsilon>0$ small enough such that
$l^{z}_{|[-1,-1+2\varepsilon]\cup[1-2\varepsilon,1]}$ is constant. Consider
also a small Weinstein chart
$e:({\mathbb{B}}^{3},\xi_{\operatorname{std}})\rightarrow(M,\xi)$ around $l$
such that

$e(x,0,0)=l(x).$

Define the family of smooth charts $e^{z}=\varphi^{z}\circ e$. Now fix the
family of smooth embeddings of
$s^{z}=e_{|\partial{\mathbb{B}}^{3}(1-\varepsilon)}^{z}:{\mathbb{S}}^{2}\rightarrow
M$. Further denote the base point by $s=s^{N}$. Observe that $s^{z}$ defines
an element in $\pi_{k}({\mathfrak{Emb}}^{s}({\mathbb{S}}^{2},M))$.

Apply Theorem 3.4.1 to obtain a homotopic $C^{0}$-close family of standard
spheres $\tilde{s}^{z}$. Now, apply isotopy extension to deform the family
$\varphi^{z}$ into a new family $\varphi^{z}_{\frac{1}{2}}$ such that
$\varphi^{z}_{\frac{1}{2}}\circ s=\tilde{s}^{z}$. By an application of Theorem
3.1.2 and Gray Stability one can further assume that
$\varphi^{z}_{\frac{1}{2}}$ is already a contactomorphism over
$e({\mathbb{B}}^{3}(1-\varepsilon))$. Finally, it remains to deform
$\varphi^{z}_{\frac{1}{2}}$ in such a way that is a contactomorphism over
${\mathcal{O}p}(l(-1,-1+\varepsilon)\cup l(1-\varepsilon,1))$. It follows from
the fact that the space of smooth long unknots is contractible that one may
further assume $\varphi^{z}_{\frac{1}{2}}$ preserves $l(-1,-1+\varepsilon)\cup
l(1-\varepsilon,1)$ point wisely but, in order to be a contactomorphism over
${\mathcal{O}p}(l(-1,-1+\varepsilon)\cup l(1-\varepsilon,1))$ it should
preserve an open neighbourhood of these Legendrian arcs (see Lemma 2.4.5);
i.e. the normal framing of the arcs. This is not true for general
diffeomorphisms and here one should use the formal structure over
$\varphi^{z}$ in order to ensure this property.555Observe that $\varphi^{z}$
and $\varphi^{z}_{\frac{1}{2}}$ preserves $l(-1,-1+\varepsilon)\cup
l(1-\varepsilon,1)$ but the homotopy $\varphi^{z}_{t}$ does not. There is no
reason for the normal framing to be preserved. The formal contact structure is
used exactly here: to make sure that the normal framing is preserved through
the deformation. Use this observation to homotope $\varphi^{z}_{\frac{1}{2}}$
into a new family $\varphi^{z}_{1}$ that is a contactomorphism over
${\mathcal{O}p}(l)$. ∎

###  4.5. The failure of the non-relative case.

We explain with an explicit example why we are not able to improve Theorem
1.0.4 to the general contactomorphisms group without considering to fix a
point in the manifold. Consider a Legendrian $1$–cell $l$ and take coordinates
$({\mathbb{R}}^{3},\xi_{\operatorname{std}}=\ker(dz-ydx)$) in a neighbourhood
of $l$ such that $l:[0,1]\rightarrow{\mathbb{R}}^{3},t\mapsto(t,0,0)$. Recall
that $({\mathbb{R}}^{3},\xi_{\operatorname{std}})$ is the contactization of
$({\mathbb{R}}^{2}(x,y),\omega_{\operatorname{std}}=d(-ydx))$ and, thus, every
area preserving diffeomorphism of ${\mathbb{R}}^{2}$ can be lifted to a
contactomorphism of $({\mathbb{R}}^{3},\xi_{\operatorname{std}})$. Consider
the annulus

$A=\\{p\in{\mathbb{R}}^{2}:\frac{1}{4}\leq||p||\leq\frac{1}{2}\\}$

and the contactomorphism
$\varphi\in\operatorname{Cont}({\mathbb{R}}^{3},\xi_{\operatorname{std}})$
obtained as the lift of a the square of a positive Dehn Twist in $A$. This
lift can be constructed in such a way that

$\varphi\in\operatorname{Cont}({\mathbb{R}}^{3},\xi_{\operatorname{std}};\operatorname{rel}\\{l(0),l(1)\\}).$

It is easy to check that $\varphi$ is isotopic to the identity in
$\operatorname{Diff}({\mathbb{R}}^{3};\operatorname{rel}\\{l(0),l(1)\\})$ but
it is clearly not contact isotopic to the identity: The rotation number of $l$
and $\varphi(l)$ are not equal; i.e. both Legendrians are not isotopic
relative to the end points. See Figure 7. Observe that the failure comes from
the fact that the long Legendrian unknots $l$ and $\varphi(l)$ are not
formally isotopic.

Figure 7. Deformation in Lagrangian projection of a $0$-cell back to the
initial position.

####  4.5.1. Geiges-Gonzalo, Bourgeois, Ghiginni example

Figure 8. The two Ghiggini legendrians. Figure 9. Required move to put the
$0$-cells back in position.

Fix the second Giroux-Kanda contact structure $\xi_{2}$ in ${\mathbb{T}}^{3}$,
i.e. a 2:1 cover of the standard one, alternatively the unique tight contact
structure on ${\mathbb{T}}^{3}$ whose Giroux torsion is $2$. The contact
structure is given by $\ker((\cos 4\pi\theta)dx+(\sin 4\pi\theta)dy)$ with
$(\theta,x,y)\in{\mathbb{T}}^{3}$.

There are two Legendrian knots that have been studied by Ghiggini [Ghi06].
They are given by $\gamma_{0}(t)=(0,0,t)$ and $\gamma_{1}(t)=(1/2,0,t)$.
Ghiggini has proven that they are smoothly isotopic (even formal Legendrian