On the contact mapping class group of Legendrian circle bundles

Abstract

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of non-positive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. This connectedness question is also studied for itself with a number of (hopefully instructive) examples.

Introduction

In this paper, we study contact transformations of 3-manifolds which are circle bundles equipped with contact structures tangent to the fibers. The main example of such a manifold is the unit cotangent bundle $V=T_{1}^{\ast }S$ of a surface $S$ , endowed with its canonical contact structure $\unicode[STIX]{x1D709}$ : this contact manifold is also called the manifold of cooriented contact elements over $S$ . Other examples are obtained as follows: for any positive integer $d$ dividing $|2g-2|$ where $g$ is the genus of $S$ , the manifold $V$ admits a $d$ -fold fibered cyclic cover $V_{d}$ and the pullback $\unicode[STIX]{x1D709}_{d}$ of $\unicode[STIX]{x1D709}$ on $V_{d}$ is a contact structure tangent to the fibers of $V_{d}$ over $S$ . It is a nice and easy observation that all Legendrian circle bundles are of this form (see [Lut83, p. 179]).

Our goal here is to determine the contact mapping class group of $(V_{d},\unicode[STIX]{x1D709}_{d})$ , namely the group $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d})$ , where ${\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d})$ denotes the group of contact transformations of $(V_{d},\unicode[STIX]{x1D709}_{d})$ (diffeomorphisms preserving the contact structure with its coorientation). This group has an obvious homomorphism to the usual (smooth) mapping class group $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d})$ (where ${\mathcal{D}}(V_{d})$ consists of all diffeomorphisms of $V_{d}$ ) which has been computed in [Wal67]. By standard fibration results (see § 1), the kernel of this homomorphism is tightly related to the fundamental group of the isotopy class of $\unicode[STIX]{x1D709}_{d}$ , i.e. the connected component of $\unicode[STIX]{x1D709}_{d}$ in the space ${\mathcal{C}}{\mathcal{S}}(V_{d})$ of all contact structures on $V_{d}$ .

Our main result is the following theorem, in which $V_{d}$ is endowed with any principal circle bundle structure inherited from one on $V=T_{1}^{\ast }S$ .

Theorem. (Theorem 2.5, Corollary 2.6, and Theorem 2.9) Let $S$ be a closed, connected, orientable surface of genus $g\geqslant 1$ and $d$ a positive integer dividing $2g-2$ . Denote by $R_{t}:V_{d}\rightarrow V_{d}$ the action of $2\unicode[STIX]{x1D70B}t\in \mathbb{R}/2\unicode[STIX]{x1D70B}\mathbb{Z}$ by rotation along the fibers. Then:

1. – the fundamental group $\unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}}(V_{d}),\unicode[STIX]{x1D709}_{d})$ is infinite cyclic and generated by the loop $(R_{t})_{\ast }\unicode[STIX]{x1D709}_{d}$ , $t\in [0,1/d]$ ;

2. – the kernel of the natural homomorphism

   $$\begin{eqnarray}\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d})\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d})\end{eqnarray}$$
   is the cyclic group $\mathbb{Z}/d\mathbb{Z}$ spanned by the contact mapping classes of the deck transformations of $V_{d}$ over $T_{1}^{\ast }S$ .

In the torus case, $g=1$ , Geiges and Gonzalo Pérez proved in [GG04, Theorem 2.1], also using topological methods, that the loop appearing in the previous theorem generates an infinite cyclic group in $\unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}},\unicode[STIX]{x1D709}_{d})$ (but they did not prove that this cyclic subgroup is the full group). Bourgeois reproved this using contact homology in [Bou06, Proposition 2]. Then Geiges and Klukas in [GK14] proved the theorem when $g=1$ and $d=1$ .

As a direct consequence of the theorem above, we obtain the following result.

Corollary. (Corollary 2.8) Let $S$ be a closed orientable surface of genus $g\geqslant 2$ . Then the natural homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(S)\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(T_{1}^{\ast }S;\unicode[STIX]{x1D709})\end{eqnarray}$$

induced by the differential is an isomorphism.

This corollary is stated as [Gir01b, Theorem 1] but the ‘proof’ given there contains a mistake. See § 3.1 for a detailed erratum and § 3.2 for several related examples.

In the case $g=1$ , each manifold $V_{d}$ is diffeomorphic to $\mathbb{T}^{3}=(\mathbb{R}/\mathbb{Z})^{3}$ fibering over $S=\mathbb{T}^{2}$ by the projection $(x,y,z)\mapsto (x,y)$ , and its contact structure $\unicode[STIX]{x1D709}_{d}$ can be defined by

$$\begin{eqnarray}\cos (2d\unicode[STIX]{x1D70B}z)\,dx-\sin (2d\unicode[STIX]{x1D70B}z)\,dy=0,\quad x,y,z\in \mathbb{R}/\mathbb{Z}.\end{eqnarray}$$

Then the results of [Gir94, Gir99] about the so-called ‘pre-Lagrangian tori’ readily imply that the image of the obvious homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{d})\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(\mathbb{T}^{3})=\text{SL}_{3}(\mathbb{Z})\end{eqnarray}$$

is the subgroup $\unicode[STIX]{x1D6F1}$ of transformations preserving $\mathbb{Z}^{2}\times \{0\}\subset \mathbb{Z}^{3}$ (see also [EP94]). Therefore, we have the following result.

Corollary. (Theorem 2.9) The induced homomorphism $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{1})\rightarrow \unicode[STIX]{x1D6F1}$ is an isomorphism.

Finally, for $g=0$ , an unpublished result of Fraser shows that the contact transformation group of the standard projective $3$ -space (namely, the unit cotangent bundle of the $2$ -sphere) is connected. This completes the list of contact mapping class groups for unit cotangent bundles of closed orientable surfaces.

1 Natural fibrations in contact topology

For any compact manifold $V$ with (possibly empty) boundary, we denote by ${\mathcal{D}}(V,\unicode[STIX]{x2202}V)$ the group of diffeomorphisms of $V$ relative to a neighborhood of the boundary. When the boundary of $V$ is empty, we sometimes drop $\unicode[STIX]{x2202}V$ from our notation.

Lemma 1.1. Let $(V,\unicode[STIX]{x1D709})$ be a compact contact manifold. The natural map

$$\begin{eqnarray}{\mathcal{D}}(V,\unicode[STIX]{x2202}V)\rightarrow {\mathcal{D}}(V,\unicode[STIX]{x2202}V)\boldsymbol{\cdot }\unicode[STIX]{x1D709},\quad \unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D719}_{\ast }\unicode[STIX]{x1D709},\end{eqnarray}$$

is a locally trivial fibration whose fiber is the contact transformation group ${\mathcal{D}}(V,\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709})$ .

Proof. By the classical Cerf–Palais fibration criterion (see [Cer91, Lemma 2, p. 240, § 0.4.4] or [Pal60, Theorem A]), it suffices to show that the above map admits a continuous local section near every point $\unicode[STIX]{x1D709}_{0}\in {\mathcal{D}}(V,\unicode[STIX]{x2202}V)\boldsymbol{\cdot }\unicode[STIX]{x1D709}$ . Choose a vector field $\unicode[STIX]{x1D708}$ transverse to $\unicode[STIX]{x1D709}_{0}$ and observe that $\unicode[STIX]{x1D709}_{0}$ has a convex open neighborhood ${\mathcal{U}}$ in ${\mathcal{D}}(V,\unicode[STIX]{x2202}V)\boldsymbol{\cdot }\unicode[STIX]{x1D709}$ which consists of contact structures transverse to $\unicode[STIX]{x1D708}$ . Then Gray’s theorem associates to any contact structure $\unicode[STIX]{x1D709}_{1}\in {\mathcal{U}}$ an isotopy $\unicode[STIX]{x1D719}_{t}\in {\mathcal{D}}(V,\unicode[STIX]{x2202}V)$ , $t\in [0,1]$ , such that $\unicode[STIX]{x1D719}_{0}=\text{Id}$ and $\unicode[STIX]{x1D719}_{t\ast }\unicode[STIX]{x1D709}_{0}=(1-t)\unicode[STIX]{x1D709}_{0}+t\unicode[STIX]{x1D709}_{1}$ for all $t\in [0,1]$ . Moreover, one can easily arrange that this isotopy varies continuously with $\unicode[STIX]{x1D709}_{1}$ . Indeed, it is uniquely determined by a smooth path $\unicode[STIX]{x1D70E}_{t}$ of sections $V\rightarrow TV/\unicode[STIX]{x1D709}_{t}$ and, if this path depends continuously on $\unicode[STIX]{x1D709}_{1}$ (if $\unicode[STIX]{x1D70E}_{t}$ is identically $0$ , for instance), then so does the corresponding isotopy, and the map $\unicode[STIX]{x1D709}_{1}\mapsto \unicode[STIX]{x1D719}_{1}$ gives the required continuous section.◻

Assume from now on that the contact manifold $(V,\unicode[STIX]{x1D709})$ has dimension three, and let $F$ be a compact orientable surface properly embedded in $V$ . Recall that the characteristic foliation $\unicode[STIX]{x1D709}F$ of $F$ in $(V,\unicode[STIX]{x1D709})$ is the (singular) foliation spanned by the line field $\unicode[STIX]{x1D709}\cap TF$ (the singularities are the points where $\unicode[STIX]{x1D709}=TF$ ). We denote by:

1. – ${\mathcal{P}}(F,V)$ the space of proper embeddings $F\rightarrow V$ which coincide with the inclusion $\unicode[STIX]{x1D704}:F\rightarrow V$ near $\unicode[STIX]{x2202}F$ ;

2. – ${\mathcal{P}}_{\!\!o}(F,V)\subset {\mathcal{P}}(F,V)$ the connected component of the inclusion $\unicode[STIX]{x1D704}$ ;

3. – ${\mathcal{P}}(F,V;\unicode[STIX]{x1D709})\subset {\mathcal{P}}(F,V)$ the subspace of embeddings $\unicode[STIX]{x1D713}$ which induce the same characteristic foliation as the inclusion, i.e. satisfy $\unicode[STIX]{x1D709}\,\unicode[STIX]{x1D713}(F)=\unicode[STIX]{x1D713}_{\ast }(\unicode[STIX]{x1D709}F)$ ;

4. – ${\mathcal{P}}_{\!\!o}(F,V;\unicode[STIX]{x1D709})$ the intersection ${\mathcal{P}}_{\!\!o}(F,V)\cap {\mathcal{P}}(F,V;\unicode[STIX]{x1D709})$ .

The same standard tools as in the proof of Lemma 1.1 give the following result.

Lemma 1.2. Let $(V,\unicode[STIX]{x1D709})$ be a compact contact manifold of dimension three. For every properly embedded surface $F\subset V$ , the restriction map

$$\begin{eqnarray}{\mathcal{D}}(V,\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709})\rightarrow {\mathcal{P}}(F,V;\unicode[STIX]{x1D709}),\quad \unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D719}|_{F},\end{eqnarray}$$

is a locally trivial fibration over its image.

Proof. Each embedding $\unicode[STIX]{x1D713}_{0}\in {\mathcal{P}}(F,V;\unicode[STIX]{x1D709})$ which lies in the image is also in the image of the restriction map

$$\begin{eqnarray}{\mathcal{D}}(V,\unicode[STIX]{x2202}V)\rightarrow {\mathcal{P}}(F,V),\quad \unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D719}|_{F},\end{eqnarray}$$

which is a locally trivial fibration by the Cerf–Palais fibration theorem. As a result, there exists a neighborhood ${\mathcal{V}}$ of $\unicode[STIX]{x1D713}_{0}$ in ${\mathcal{P}}(F,V;\unicode[STIX]{x1D709})$ , and a continuous extension map ${\mathcal{V}}\rightarrow {\mathcal{D}}(V,\unicode[STIX]{x2202}V)$ which associates to every embedding $\unicode[STIX]{x1D713}_{1}\in {\mathcal{V}}$ a diffeomorphism $\unicode[STIX]{x1D719}_{1}\in {\mathcal{D}}(V,\unicode[STIX]{x2202}V)$ such that $\unicode[STIX]{x1D713}_{1}=\unicode[STIX]{x1D719}_{1}|_{F}$ . Using Gray’s theorem, and the fact that embeddings in ${\mathcal{V}}$ induce the same characteristic foliation, it is easy to correct this extension map so that it takes values in ${\mathcal{D}}(V,\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709})$ . We conclude applying again the Cerf–Palais fibration criterion.◻

Remark 1.3. The above lemma is a typical result where it is useful to work relatively to a neighborhood of the boundary and not just to the boundary itself. Indeed, any diffeomorphism relative to both $\unicode[STIX]{x2202}V$ and a properly embedded surface $F$ is tangent to the identity along $\unicode[STIX]{x2202}F$ , and so the fibration property fails in this case. However, since the inclusion of ${\mathcal{D}}(V,\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709})$ into the group of contact transformations relative to the boundary is a homotopy equivalence, this does not matter.

We now recall how the theory of $\unicode[STIX]{x1D709}$ -convex surfaces can be used to study the homotopy type of ${\mathcal{P}}(F,V;\unicode[STIX]{x1D709})$ (see [Gir91, Gir01b]). Let $F$ be a compact orientable surface properly embedded in $(V,\unicode[STIX]{x1D709})$ with (possibly empty) Legendrian boundary; $F$ is $\unicode[STIX]{x1D709}$ -convex if it admits a homogeneous neighborhood, namely a product neighborhood

$$\begin{eqnarray}U:=F\times \mathbb{R}\supset F=F\times \{0\}\quad \text{with}~\unicode[STIX]{x2202}U=\unicode[STIX]{x2202}F\times \mathbb{R}\subset \unicode[STIX]{x2202}V,\end{eqnarray}$$

in which the vector field $\unicode[STIX]{x2202}_{t}$ , $t\in \mathbb{R}$ , preserves $\unicode[STIX]{x1D709}$ . The points $p\in F$ where $\unicode[STIX]{x2202}_{t}(p)\in \unicode[STIX]{x1D709}$ then form a multi-curve $\unicode[STIX]{x1D6E4}$ called the dividing set of $F$ associated with $U$ . This curve depends on $U$ and its product structure, but its isotopy class does not, and is uniquely determined by the foliation $\unicode[STIX]{x1D709}F$ : specifically, $\unicode[STIX]{x1D6E4}$ is the unique multi-curve (up to isotopy) which avoids the singularities of $\unicode[STIX]{x1D709}F$ , is transverse to $\unicode[STIX]{x1D709}F$ and divides $F$ into regions where the dynamics of $\unicode[STIX]{x1D709}F$ is alternatively expanding and contracting (see [Gir91, Gir01b] for more details). It follows that the curves dividing a given singular foliation on a surface, in the above sense, form a contractible space. Among them, the dividing sets associated with all possible homogeneous neighborhoods $U$ of $F$ are those intersecting $\unicode[STIX]{x2202}F$ at the points where $\unicode[STIX]{x1D709}$ is tangent to $\unicode[STIX]{x2202}V$ . Moreover, we have the following proposition (see [Gir01b, Lemmas 6 and 7]).

Proposition 1.4. Let $F$ be a $\unicode[STIX]{x1D709}$ -convex surface, $U$ a homogeneous neighborhood, and $\unicode[STIX]{x1D6E4}$ the associated dividing set.

1. (a) The space $\mathscr{F}(F;\unicode[STIX]{x1D6E4})$ of singular foliations on $F$ which are tangent to $\unicode[STIX]{x2202}F$ and admit $\unicode[STIX]{x1D6E4}_{U}$ as a dividing set is an open contractible neighborhood of $\unicode[STIX]{x1D709}F$ in the space of all singular foliations on $F$ .

2. (b) There exists a continuous map $\mathscr{F}(F;\unicode[STIX]{x1D6E4})\rightarrow {\mathcal{P}}(F,V)$ , $\unicode[STIX]{x1D70E}\mapsto \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}}$ , with the following properties:

   1. (i) $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D709}F}$ is the inclusion $F\rightarrow V$ ;

   2. (ii) $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}}(F)$ is contained in $U=F\times \mathbb{R}$ and transverse to the contact vector field $\unicode[STIX]{x2202}_{t}$ for all $\unicode[STIX]{x1D70E}\in \mathscr{F}(F;\unicode[STIX]{x1D6E4})$ ;

   3. (iii) $\unicode[STIX]{x1D709}\,\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}}(F)=\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D70E})$ for all $\unicode[STIX]{x1D70E}\in \mathscr{F}(F;\unicode[STIX]{x1D6E4})$ .

3. (c) Let ${\mathcal{P}}(F,V;\unicode[STIX]{x1D6E4})$ denote the space of embeddings $\unicode[STIX]{x1D713}\in {\mathcal{P}}(F,V)$ such that $\unicode[STIX]{x1D713}(F)$ is $\unicode[STIX]{x1D709}$ -convex with dividing set $\unicode[STIX]{x1D713}(\unicode[STIX]{x1D6E4})$ . Then the inclusion ${\mathcal{P}}(F,V;\unicode[STIX]{x1D709})\rightarrow {\mathcal{P}}(F,V;\unicode[STIX]{x1D6E4})$ is a homotopy equivalence.

We will also need the following result which shows that the homotopy type of ${\mathcal{D}}(V,\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709})$ is locally constant when $\unicode[STIX]{x2202}V$ is $\unicode[STIX]{x1D709}$ -convex (see [Gir01b, Proposition 8].

Proposition 1.5. Let $V$ be a compact $3$ -manifold,  $\unicode[STIX]{x1D6E5}$ a multi-curve on $\unicode[STIX]{x2202}V$ , and ${\mathcal{C}}{\mathcal{S}}(V,\unicode[STIX]{x1D6E5})$ the space of contact structures $\unicode[STIX]{x1D709}$ on $V$ for which $\unicode[STIX]{x2202}V$ is $\unicode[STIX]{x1D709}$ -convex with dividing set $\unicode[STIX]{x1D6E5}$ . For $\unicode[STIX]{x1D709}\in {\mathcal{C}}{\mathcal{S}}(V,\unicode[STIX]{x1D6E5})$ , the homotopy type of ${\mathcal{D}}(V,\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709})$ depends only on the connected component of ${\mathcal{C}}{\mathcal{S}}(V;\unicode[STIX]{x1D6E5})$ containing $\unicode[STIX]{x1D709}$ .

2 Legendrian circle bundles over surfaces

2.1 The general case

In this section, we consider a compact oriented surface $S$ which is neither a sphere nor a torus. The torus case will be treated in § 2.2. Actually, using results of [Mas08], the following discussion can be carried over to orbifolds.

As in the introduction, $(V_{d},\unicode[STIX]{x1D709}_{d})$ denotes the $d$ -fold fibered cyclic cover of the unit cotangent bundle $V:=V_{1}=T_{1}^{\ast }S$ , equipped with the pullback of the canonical contact structure $\unicode[STIX]{x1D709}$ of $V$ .

In any trivialization of $\unicode[STIX]{x1D70B}:V_{d}\rightarrow S$ over a subsurface $R\subset S$ with non-empty boundary, the contact structure $\unicode[STIX]{x1D709}_{d}$ can be described as follows: if $J$ denotes the complex structure on $S$ induced by our choice of a principal bundle structure on $T_{1}^{\ast }S$ , then the restriction of $\unicode[STIX]{x1D709}_{d}$ to $V_{d}|_{R}\simeq R\times \mathbb{S}^{1}$ has the form

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{\unicode[STIX]{x1D706},d}=\ker (\cos (d\unicode[STIX]{x1D703})\unicode[STIX]{x1D706}+\sin (d\unicode[STIX]{x1D703})\unicode[STIX]{x1D706}\circ J),\quad \unicode[STIX]{x1D703}\in \mathbb{S}^{1},\end{eqnarray}$$

where $\unicode[STIX]{x1D706}$ is a non-singular 1-form on $R$ . In practice, we will take $R$ equal to $S$ if $S$ has non-empty boundary, and to $S$ with an open disk removed if $S$ is closed.

Now observe that the preimage $F:=\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D6FE})$ of any properly embedded curve $\unicode[STIX]{x1D6FE}$ in $S$ is a $\unicode[STIX]{x1D709}_{d}$ -convex surface in $V_{d}$ . Indeed, any vector field $X$ in $S$ transverse to $\unicode[STIX]{x1D6FE}$ (and tangent to $\unicode[STIX]{x2202}S$ ) lifts to a contact vector field $\bar{X}$ transverse to $F$ (and tangent to $\unicode[STIX]{x2202}V_{d}$ ). The dividing set of $\unicode[STIX]{x1D709}_{d}F$ associated with $\bar{X}$ is the set of points in $F$ where $\unicode[STIX]{x1D709}_{d}$ projects down (by the differential of $\unicode[STIX]{x1D70B}$ ) to the line spanned by $X$ .

If $\unicode[STIX]{x1D6FE}$ is contained in the subsurface $R$ , the trivialization of $V_{d}|_{R}$ induces a diffeomorphism $F=\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D6FE})\simeq \unicode[STIX]{x1D6FE}\times \mathbb{S}^{1}$ . Then the dividing set, provided all its components are consistently oriented (as parallel curves), represents the homology class $2(d,x-1)\in \mathbb{Z}^{2}=H_{1}(\unicode[STIX]{x1D6FE}\times \mathbb{S}^{1})$ , where $x$ denotes the index of $\unicode[STIX]{x1D706}$ along $\unicode[STIX]{x1D6FE}$ .

The following proposition can be proved using [Gir00, Lemma 4.7] (a special case of the semi-local Bennequin inequality proved later as [Gir01a, Proposition 4.10]) exactly as in [Gir01a, Lemma 3.9] which dealt with circle bundles without boundary.

Proposition 2.1. Let $F$ be a torus fibered over a homotopically essential circle in $S$ , and $\unicode[STIX]{x1D6E4}$ a dividing set for $\unicode[STIX]{x1D709}_{d}F$ . For any isotopy $\unicode[STIX]{x1D711}$ such that $\unicode[STIX]{x1D711}_{1}(F)$ is also $\unicode[STIX]{x1D709}_{d}$ -convex, the foliation $\unicode[STIX]{x1D709}_{d}\unicode[STIX]{x1D711}_{1}(F)$ is divided by a collection of curves isotopic to the components of $\unicode[STIX]{x1D711}_{1}(\unicode[STIX]{x1D6E4})$ .

We now turn to spaces of embeddings of surfaces. The following lemma is useful to prove the existence of contact transformations which are smoothly but not contact isotopic to the identity.

Lemma 2.2. Let $T$ be a fibered torus over a homotopically non-trivial circle $C$ in $S$ , and $i:T\rightarrow V_{d}$ the inclusion map. Let $R_{t}$ be the action of $e^{2i\unicode[STIX]{x1D70B}t}$ on $V_{d}$ . For any non-zero integer $k$ in $\mathbb{Z}$ , the path $\unicode[STIX]{x1D6FE}_{k}:[0,1]\rightarrow {\mathcal{P}}_{\!\!o}(T,V_{d})$ defined by $\unicode[STIX]{x1D6FE}_{k}(t)=R_{kt/d}\circ i$ is non-trivial in $\unicode[STIX]{x1D70B}_{1}({\mathcal{P}}_{\!\!o}(T,V_{d}),{\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d}))$ .

For any integer $k$ between $1$ and $d-1$ , the action of $R_{k/d}$ on $\unicode[STIX]{x1D70B}_{0}({\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d}))$ is non-trivial.

The above lemma will be reduced to the following statement.

Proposition 2.3 [Ghi06, Proposition 7.1].

In $(\mathbb{T}^{2}\times \mathbb{R},\ker (\cos (2n\unicode[STIX]{x1D70B}z)\,dx-\sin (2n\unicode[STIX]{x1D70B}z)\,dy))$ , the Legendrian circles $\{0\}\times \mathbb{S}^{1}\times \{0\}$ and $\{0\}\times \mathbb{S}^{1}\times \{k\}$ are not contact isotopic for any $k\neq 0$ .

Alternatively, one could use the stronger result due to Eliashberg et al. [EHS95] saying that, in $(\mathbb{T}^{3},\ker (\cos (2n\unicode[STIX]{x1D70B}z)\,dx-\sin (2n\unicode[STIX]{x1D70B}z)\,dy))$ , the Legendrian circle $\{0\}\times \mathbb{S}^{1}\times \{0\}$ cannot be displaced from the pre-Lagrangian torus $\mathbb{S}^{1}\times \mathbb{S}^{1}\times \{0\}$ by a contact isotopy. However, this result uses holomorphic curves in symplectizations so it has a different flavor from the techniques we use in this paper.

Proof of Lemma 2.2.

We first explain how to reduce the first statement to a statement in a thickened torus. Suppose for contradiction that there is a path $j$ in ${\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d})$ from the inclusion to $\unicode[STIX]{x1D6FE}_{k}(1)$ . The path lifting property of the fibration ${\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d})\rightarrow {\mathcal{P}}(T,V_{d};\unicode[STIX]{x1D709}_{d})$ of Lemma 1.2 gives a contact isotopy $\unicode[STIX]{x1D711}_{t}$ such that $\unicode[STIX]{x1D6FE}_{k}(t)=\unicode[STIX]{x1D711}_{t}\circ i$ . Let $p:{\hat{S}}\rightarrow S$ be the covering map associated to the subgroup generated by $C$ in $\unicode[STIX]{x1D70B}_{1}(S)$ , and let $\hat{V}$ be the induced circle bundle over ${\hat{S}}$ . We denote by $\hat{T}$ the compact component of $p^{-1}(T)$ . The contact isotopy $\unicode[STIX]{x1D711}$ lifts to a contact isotopy $\hat{\unicode[STIX]{x1D711}}$ for the induced contact structure $p^{\ast }\unicode[STIX]{x1D709}_{d}$ . The interior of ${\hat{S}}$ is an open annulus, and contains a closed sub-annulus $A$ such that $\hat{\unicode[STIX]{x1D711}}_{t}(\hat{T})$ stays above $A$ for all $t$ . One can then cut off $\hat{\unicode[STIX]{x1D711}}_{t}$ , using Libermann’s theorem from [Lib59], to get a contact isotopy with support in $A\times \mathbb{S}^{1}$ . In $H_{1}(\hat{T},\mathbb{Z})$ , we consider a $\mathbb{Z}$ -basis $(S,F)$ , where $F$ is the homology class of fibers. The torus $\hat{T}$ has circles of singularities which can be oriented consistently to get a total homology class $2(dS+mF)$ , where $m$ is an unknown integer. After pulling back everything under a $d$ -fold covering map from $A$ to itself, we can assume there are exactly $2d$ circles of singularities. The circle bundle over $A$ then embeds into $\mathbb{T}^{3}$ equipped with $\ker (\cos (2d\unicode[STIX]{x1D70B}z)\,dx-\sin (2d\unicode[STIX]{x1D70B}z)\,dy)$ , so it is sufficient to get a contradiction there. Note that we do not make any claim concerning how the circle bundle structure embeds inside $\mathbb{T}^{3}$ and we will not use it.

We now consider the covering map $\mathbb{T}^{2}\times \mathbb{R}\rightarrow \mathbb{T}^{3}$ sending $(x,y,s)$ to $(x,y,s~\mod ~\mathbb{Z})$ . The contact isotopy $\hat{\unicode[STIX]{x1D711}}$ lifts to a contact isotopy contradicting Proposition 2.3.

The statement about $\unicode[STIX]{x1D70B}_{0}({\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d}))$ follows immediately from what we proved and the long exact sequence of the pair $({\mathcal{P}}_{\!\!o}(T,V_{d}),{\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d}))$ , since the path corresponding to $k$ between $1$ and $d-1$ does not come from a loop in ${\mathcal{P}}_{\!\!o}(T,V_{d})$ .◻

Proposition 2.4. If $T$ is a fibered torus over a non-separating embedded circle in $S$ , then the group of deck transformations of $V_{d}\rightarrow V$ acts freely and transitively on $\unicode[STIX]{x1D70B}_{0}({\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d}))$ .If $A$ is a fibered annulus over a non-separating properly embedded arc in $S$ , then ${\mathcal{P}}_{\!\!o}(A,V_{d};\unicode[STIX]{x1D709}_{d})$ is connected.

The following proof will need one more technical ingredient from the study of contact structures on circle bundles: the twisting number. Fibers in $V_{d}$ have a canonical framing coming from vector fields along the fiber which project to some constant vector on the base. If $L$ is any Legendrian curve isotopic to a fiber, we call twisting number of $L$ the number $t(L)$ of turns made by $\unicode[STIX]{x1D709}_{d}$ along $L$ compared to the canonical framing transported by isotopy from some fiber, see [Gir01a, p. 227] for further discussion. As explained in [Gir01a, Lemma 3.6], it follows from Bennequin’s inequality in $\mathbb{R}^{3}$ that $t(L)\leqslant -d$ for all $L$ .

Proof. We first prove connectedness of ${\mathcal{P}}_{\!\!o}(A,V_{d};\unicode[STIX]{x1D709}_{d})$ . Let $(j_{t})_{t\in [0,1]}$ be an isotopy of embeddings of $A$ which coincides with the inclusion map on a neighborhood of $\unicode[STIX]{x2202}A$ . Let $\unicode[STIX]{x1D6E4}$ be a dividing set on $A$ associated to some homogeneous neighborhood. According to Proposition 1.4(c), we only need to prove that $j$ is homotopic to a path in ${\mathcal{P}}_{\!\!o}(A,V_{d};\unicode[STIX]{x1D6E4})$ .

We use Colin’s discretization technique [Col97], which relies on the following observation. We can find times $t_{0}=0<t_{1}<\cdots <t_{k}=1$ such that, for $t$ in $[t_{i},t_{i+1}]$ , the annuli $j_{t}(A)$ are all contained in some pinched product: $A_{i}\times [0,1]$ with $\{x\}\times [0,1]$ collapsed to a point for $x$ in a neighborhood of $\unicode[STIX]{x2202}A_{i}$ . The sub-path $j_{t}$ , $t\in [t_{i},t_{i+1}]$ is then homotopic, with fixed end points, to the concatenation of two paths whose common extremity has image $A_{i}\times \{0\}$ . In addition, genericity of $\unicode[STIX]{x1D709}_{d}$ -convex surfaces allows us to assume $A_{i}\times \{0\}$ is $\unicode[STIX]{x1D709}_{d}$ -convex. So we have replaced $j$ by the concatenation of $2k$ paths of embeddings sweeping out pinched products. We can assume there is only one such path, and the general case follows by induction.

The classification of tight contact structures on solid tori [Gir00, Hon00] guarantees that, in this situation, $j$ is homotopic to an isotopy in ${\mathcal{P}}_{\!\!o}(A,V_{d};\unicode[STIX]{x1D6E4})$ as soon as $A^{\prime }=j_{1}(A)$ is divided by a curve $\unicode[STIX]{x1D6E4}^{\prime }$ isotopic to $j_{1}(\unicode[STIX]{x1D6E4})$ . So we prove that fact.

We first remark that both $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6E4}^{\prime }$ are made of $2d$ traversing curves, because otherwise we could use the realization lemma (Proposition 1.4(b)) to produce a Legendrian circle $L$ isotopic to the fibers with twisting number $t(L)>-d$ . We now need two separate arguments, depending on the value of $d$ .

Suppose first that $d$ is greater than one. Let $A^{\prime \prime }$ and $A^{\prime \prime \prime }$ be annuli isotopic to $A^{\prime }$ through $\unicode[STIX]{x1D709}_{d}$ -convex surfaces and such that the annuli $A$ , $A^{\prime }$ , $A^{\prime \prime }$ , $A^{\prime \prime \prime }$ pairwise bound pinched products, and there is an arc going from $A$ to $A^{\prime \prime \prime }$ in the pinched product they bound and meeting $A^{\prime }$ and then $A^{\prime \prime }$ in its interior. Near their boundary, all of these annuli are fibered over arcs in $S$ . Let $T\times [0,1]$ be a thickened torus with $\unicode[STIX]{x1D709}_{d}$ -convex boundary $T_{0}\sqcup T_{1}$ , such that $T_{0}$ is a smoothing of $A\cup A^{\prime \prime \prime }$ , $T_{1}$ is a smoothing of $A^{\prime }\cup A^{\prime \prime }$ , and those tori are fibered in the smoothing region. We identify the first homology groups of $T_{0}$ and $T_{1}$ using the product $T\times [0,1]$ , and fix an integer basis $(S,F)$ where $F$ is the class coming from fibers of $V$ .

If $\unicode[STIX]{x1D6E4}^{\prime }$ is not isotopic to $j_{1}(\unicode[STIX]{x1D6E4})$ , then, after orienting all dividing curves of $T_{0}$ and $T_{1}$ in the same way, their total homology class is $2(d,x_{0})$ on $T_{0}$ and $2(d,x_{1})$ on $T_{1}$ with $x_{1}\neq x_{0}$ . Pick’s formula, proved in [Pic99], ensures that the triangle with vertices $(0,0)$ , $(d,x_{0})$ and $(d,x_{1})$ in $H_{1}(T;\mathbb{R})$ contains integer points outside its vertical edge. If $(a,b)$ is such a point, then $a<d$ . The classification of tight contact structures on thickened tori then gives a $\unicode[STIX]{x1D709}_{d}$ -convex torus in $T\times [0,1]$ divided by a collection of parallel curves which can be oriented all in the same way to have total homology class $2(a,b)$ . The realization lemma gives again a Legendrian curve with twisting number $t=-a>-d$ , hence a contradiction.

We now handle the case $d=1$ . In particular $(V_{d},\unicode[STIX]{x1D709}_{d})$ is isomorphic to $(V,\unicode[STIX]{x1D709})$ . Adding 1-handles, one can easily embed $S$ into a surface $S^{\prime }$ with connected boundary so that $\unicode[STIX]{x1D709}$ extends to a contact structure tangent to the fibers of $S^{\prime }\times \mathbb{S}^{1}$ and, of course, the projection of $A$ stays non-separating in $S^{\prime }$ . So we assume $S$ has connected boundary. Let $S^{\prime \prime }$ be the surface obtained by gluing a disk along the boundary of $S$ . The description of the contact structure $\unicode[STIX]{x1D709}$ in terms of a 1-form $\unicode[STIX]{x1D706}$ on $S$ allows us to understand the characteristic foliation $\unicode[STIX]{x1D709}\unicode[STIX]{x2202}V$ in terms of the index of $\unicode[STIX]{x1D706}$ along $\unicode[STIX]{x2202}S$ . The latter is given by the Poincaré–Hopf theorem, so it is fixed. Using this information, we can embed $(V,\unicode[STIX]{x1D709})$ into the space $V^{\prime \prime }$ of contact elements of $S^{\prime \prime }$ , with its canonical contact structure. In $V^{\prime \prime }$ , one can extend $A$ and $A^{\prime }$ to isotopic non-separating tori, which coincide outside $V$ . Both tori are divided by two curves, and Proposition 2.1 guarantees those curves are isotopic. This implies that $\unicode[STIX]{x1D6E4}^{\prime }$ is isotopic to $j_{1}(\unicode[STIX]{x1D6E4})$ .

We now turn to the case of a torus $T$ fibered over a non-separating circle $C$ in $S$ . We fix a dividing set $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D709}_{d}T$ . Let $j$ be any isotopy of embeddings of $T$ in $V_{d}$ such that $T^{\prime }=j_{1}(T)$ is $\unicode[STIX]{x1D709}_{d}$ -convex. Proposition 2.1 ensures that any component of any dividing set of $\unicode[STIX]{x1D709}_{d}T^{\prime }$ is isotopic in $T^{\prime }$ to a component of $j_{1}(\unicode[STIX]{x1D6E4})$ . However, there always exist isotopies of $T$ which change the number of dividing curves and, as explained in § 3, there is no general result allowing us to get rid of them. This is where we need $C$ to be non-separating, and not only homotopically non-trivial. The idea, which was born in [Ghi05, Proposition 5.4] and further developed in [Mas08, Lemma 8.5], is to consider a fibered torus $F$ intersecting $T$ along one fiber. Then for any isotopy $\unicode[STIX]{x1D711}_{t}$ , one can discretize the movement of $F$ while constructing an isotopy of $T$ through $\unicode[STIX]{x1D709}_{d}$ -convex surfaces. There is no boundary in [Mas08], but one can check that it does not change anything here. This trick constructs a contact isotopy $\unicode[STIX]{x1D711}_{t}^{\prime }$ such that $\unicode[STIX]{x1D711}_{1}^{\prime }(T)=\unicode[STIX]{x1D711}_{1}(T)$ . Of course parametrizations do not match in general: $(\unicode[STIX]{x1D711}_{1}^{\prime })^{-1}\circ \unicode[STIX]{x1D711}_{1}$ induces a self-diffeomorphism of $T$ which may fail to be isotopic to the identity among diffeomorphisms preserving $\unicode[STIX]{x1D709}_{d}T$ . However, after composing $\unicode[STIX]{x1D711}_{1}^{\prime }$ by a deck transformation, we can assume that each circle of singularities of $T$ is globally preserved and, after an ultimate contact isotopy, we get a path in ${\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d})$ . So the deck transformations group acts transitively on $\unicode[STIX]{x1D70B}_{0}({\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d}))$ . The last part of Lemma 2.2 states that this action is also free.◻

Theorem 2.5. If $S$ is closed, then the kernel of the canonical homomorphism $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d},\unicode[STIX]{x1D709}_{d})\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d})$ is the cyclic group of deck transformations of $V_{d}$ over $V$ . If $V$ has non-empty boundary, then $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d},\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709}_{d})\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d},\unicode[STIX]{x2202}V_{d})$ is injective.

Proof. We first assume $V$ has non-empty boundary and prove that the map $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d},\unicode[STIX]{x2202}V_{d};\unicode[STIX]{x1D709}_{d})\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d},\unicode[STIX]{x2202}V_{d})$ is injective. The proof proceeds by induction on

$$\begin{eqnarray}n(S)=-2\unicode[STIX]{x1D712}(S)-\unicode[STIX]{x1D6FD}(S)=\unicode[STIX]{x1D6FD}(S)+4g(S)-4,\end{eqnarray}$$

where $\unicode[STIX]{x1D712}(S)$ and $g(S)$ are the Euler characteristic and genus of $S$ and $\unicode[STIX]{x1D6FD}(S)$ is the number of connected components of $\unicode[STIX]{x2202}S$ . So $n(S)\geqslant -3$ with equality when $S$ is a disk.

We first explain the induction step so we assume $n(S)>-3$ . Let $\unicode[STIX]{x1D711}$ be a contactomorphism of $V_{d}$ relative to some neighborhood $U$ of $\unicode[STIX]{x2202}V_{d}$ , and smoothly isotopic to the identity relative to $U$ . Let $a$ be a properly embedded non-separating arc in $S$ , and denote by $A$ the annulus fibered over $a$ , and $i:A\rightarrow V_{d}$ the inclusion map. According to Proposition 2.4, ${\mathcal{P}}_{\!\!o}(A,V_{d},\unicode[STIX]{x1D709}_{d})$ is connected. Hence, the path lifting property of the fibration ${\mathcal{D}}_{\!o}(V_{d},\unicode[STIX]{x2202}V_{d};\unicode[STIX]{x1D709}_{d})\rightarrow {\mathcal{P}}_{\!\!o}(A,V_{d};\unicode[STIX]{x1D709}_{d})$ from Lemma 1.2 implies that $\unicode[STIX]{x1D711}$ is contact isotopic to some $\unicode[STIX]{x1D711}^{\prime }$ which is relative to $A$ and $U$ . Using Remark 1.3, we can assume $\unicode[STIX]{x1D711}^{\prime }$ is relative to a neighborhood of $\unicode[STIX]{x2202}V_{d}\cup A$ which is fibered over some neighborhood $W$ of $a\cup \unicode[STIX]{x2202}S$ in $S$ . We cut $S$ along $a$ , and round the corners inside $W$ to get a subsurface $S^{\prime }\subset S$ with $n(S^{\prime })<n(S)$ . By induction hypothesis applied to $\unicode[STIX]{x1D70B}^{-1}(S^{\prime })$ , $\unicode[STIX]{x1D711}^{\prime }$ is contact isotopic to the identity so the induction step is complete.

The induction starts with the disk case, which is already explained with all details in [Gir01b, p. 345]. The idea is the same as for the induction step, but the cutting surface in the solid torus $V_{d}$ is a meridian disk. There are no such disks with Legendrian boundary in $V_{d}$ but one can use the realization lemma (Proposition 1.4) to deform $\unicode[STIX]{x1D709}_{d}$ near $\unicode[STIX]{x2202}V_{d}$ until such a disk exists. This does not change the homotopy type of ${\mathcal{D}}(V_{d},\unicode[STIX]{x2202}V_{d};\unicode[STIX]{x1D709}_{d})$ according to Proposition 1.5. A variation on Colin’s result about embedding of disks in [Col99, Theorem 3.1] then replaces Proposition 2.4, and the final isotopy is provided by Eliashberg’s result in [Eli92] that $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(B^{3},\unicode[STIX]{x2202}B^{3};\unicode[STIX]{x1D709})$ is trivial for the standard ball.

We now turn to the case where $V_{d}$ is closed. We first prove that the group of deck transformations injects into $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d})$ . Let $C$ be a non-separating circle in $S$ and $T$ the fibered torus over $C$ . Denote by $i$ the inclusion of $T$ in $V_{d}$ . Proposition 2.4 guarantees that the action of a non-trivial deck transformation $f$ on $\unicode[STIX]{x1D70B}_{0}({\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d}))$ is non-trivial. Hence, $f$ is non-trivial in $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d})$ .

We now prove surjectivity. Let $\unicode[STIX]{x1D711}$ be a contactomorphism of $V_{d}$ which is smoothly isotopic to the identity. Proposition 2.4 gives a deck transformation $f$ such that $f\circ \unicode[STIX]{x1D711}\circ i$ is isotopic to $i$ in ${\mathcal{P}}_{\!\!o}(T,V_{d};\unicode[STIX]{x1D709}_{d})$ . As above, this implies that $f\circ \unicode[STIX]{x1D711}$ is contact isotopic to a contactomorphism $\unicode[STIX]{x1D711}^{\prime }$ which is relative to an open fibered neighborhood $U$ of $T$ . The circle bundle $V_{d}\setminus U$ has non-empty boundary hence we know that $\unicode[STIX]{x1D711}^{\prime }$ is contact isotopic to identity.◻

Corollary 2.6. Assume that $V_{d}$ has empty boundary and denote by ${\mathcal{C}}{\mathcal{S}}$ the space ${\mathcal{D}}(V_{d})\boldsymbol{\cdot }\unicode[STIX]{x1D709}_{d}$ of contact structures isomorphic to $\unicode[STIX]{x1D709}_{d}$ on $V_{d}$ . Let $R_{t}$ denote the action of $e^{2i\unicode[STIX]{x1D70B}t}\in \mathbb{S}^{1}$ on $V_{d}$ . The fundamental group $\unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}},\unicode[STIX]{x1D709}_{d})$ is an infinite cyclic group generated by the loop $t\mapsto (R_{t/d})_{\ast }\unicode[STIX]{x1D709}_{d}$ .

Proof. The fibration ${\mathcal{D}}(V_{d})\rightarrow {\mathcal{C}}{\mathcal{S}}$ of Lemma 1.1 gives the exact sequence

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}({\mathcal{D}}(V_{d}),\text{Id})\rightarrow \unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}},\unicode[STIX]{x1D709}_{d})\rightarrow \unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d}))\rightarrow \unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(V_{d})).\end{eqnarray}$$

We know from [Lau74, Hat76] that $\unicode[STIX]{x1D70B}_{1}({\mathcal{D}}(V_{d}),\text{Id})$ is an infinite cyclic group generated by the loop $t\mapsto R_{t}$ , $t\in [0,1]$ . Lemma 2.2 implies that this group injects into $\unicode[STIX]{x1D70B}_{0}({\mathcal{D}}_{\!o}(V_{d}),{\mathcal{D}}_{\!o}(V_{d};\unicode[STIX]{x1D709}_{d}))\simeq \unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}},\unicode[STIX]{x1D709}_{d})$ . The result then follows from Theorem 2.5 describing the kernel of $\unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d}))\rightarrow \unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(V_{d}))$ . ◻

Before coming back to the special case of $V=V_{1}$ , we note one more general corollary of Proposition 2.1.

Lemma 2.7. A diffeomorphism of $V_{d}$ which is fibered over the identity is isotopic to a contactomorphism only if it is isotopic to the identity.

Proof. Let $f$ be a diffeomorphism of $V_{d}$ fibered over the identity of $S$ . In order to guarantee that $f$ is isotopic to the identity, it is enough to check that, for every torus $T$ fibered over a homotopically essential circle, the restriction of $f$ to $T$ preserves an isotopy class of curves which is different from the class of fibers. Assume that $f$ is isotopic to a contactomorphism. This condition means that $\unicode[STIX]{x1D709}_{d}^{\prime }=f^{\ast }\unicode[STIX]{x1D709}_{d}$ is isotopic to $\unicode[STIX]{x1D709}_{d}$ . Since $f$ is fibered, the contact structure $\unicode[STIX]{x1D709}_{d}^{\prime }$ is also tangent to the fibers. Proposition 2.1 then implies that, for each torus $T$ as above, $f$ preserves the isotopy class of dividing curves. Those dividing curves are homotopically essential, and not isotopic to fibers. Hence, $f$ is isotopic to the identity.◻

Corollary 2.8. The lifting map from $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(S)$ to $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V,\unicode[STIX]{x1D709})$ is an isomorphism.

Proof. We denote by $p$ the projection from $V$ to $S$ , and by ${\mathcal{D}}(S,\unicode[STIX]{x2202}S)$ the group of diffeomorphisms of $S$ relative to a neighborhood of $\unicode[STIX]{x2202}S$ . In the sequence of maps

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(S,\unicode[STIX]{x2202}S)\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V,\unicode[STIX]{x2202}V;\unicode[STIX]{x1D709})\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(V,\unicode[STIX]{x2202}V)\end{eqnarray}$$

the composite map is known to be injective (this follows from considerations of fundamental groups), so the first map is also injective. It remains to prove that it is surjective. Let $\unicode[STIX]{x1D711}$ be a contactomorphism. We want to prove that $\unicode[STIX]{x1D711}$ is contact isotopic to the lift of some diffeomorphism of $S$ . According to Waldhausen [Wal67, Satz 10.1], $\unicode[STIX]{x1D711}$ is smoothly isotopic to a fibered diffeomorphism $f$ : there exists an isotopy $\unicode[STIX]{x1D713}$ , and a diffeomorphism $\bar{f}$ in ${\mathcal{D}}(S,\unicode[STIX]{x2202}S)$ , such that $f=\unicode[STIX]{x1D713}_{1}\circ \unicode[STIX]{x1D711}$ and $p\circ f=\bar{f}\circ p$ . We will prove that $\unicode[STIX]{x1D711}$ is contact isotopic to the lift $D\bar{f}$ . We first note that $f\circ D(\bar{f})^{-1}$ is fibered over the identity, and is smoothly isotopic to a contactomorphism (through the path $t\mapsto \unicode[STIX]{x1D713}_{1-t}\circ \unicode[STIX]{x1D711}\circ D(\bar{f})^{-1}$ ). Lemma 2.7 then guarantees that $f\circ D(\bar{f})^{-1}$ is smoothly isotopic to the identity. Hence, $\unicode[STIX]{x1D711}$ is smoothly isotopic to $D\bar{f}$ , hence contact isotopic to $D\bar{f}$ according to Theorem 2.5.◻

2.2 The torus case

We now explain how the previous discussion can be modified to handle the case of a torus base. On $\mathbb{T}^{3}=(\mathbb{R}/\mathbb{Z})^{3}$ with coordinates $(x,y,z)$ , we set

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{d}=\ker (\cos (2d\unicode[STIX]{x1D70B}z)\,dx-\sin (2d\unicode[STIX]{x1D70B}z)\,dy).\end{eqnarray}$$

The case $d=1$ corresponds to the contact element bundle of $\mathbb{T}^{2}$ while higher values of $d$ come from self-covering maps unwrapping the fibers. We denote by $R_{t}$ the map $(x,y,z)\mapsto (x,y,z+2\unicode[STIX]{x1D70B}t)$ .

Theorem 2.9. On $(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{d})$ :

1. (1) a diffeomorphism is isotopic to a contactomorphism if and only its action on $H_{2}(\mathbb{T}^{3})$ preserves the homology class, up to sign, of the pre-Lagrangian torus $\{z=0\}$ ;

2. (2) the kernel of $\unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(\mathbb{T}^{3};\unicode[STIX]{x1D709})\rightarrow \unicode[STIX]{x1D70B}_{0}{\mathcal{D}}(\mathbb{T}^{3})$ is isomorphic to the cyclic group of order $d-1$ generated by $(x,y,z)\mapsto (x,y,z+1/d)$ ;

3. (3) the fundamental group $\unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}}(\mathbb{T}^{3}),\unicode[STIX]{x1D709}_{d})$ is an infinite cyclic group generated by the loop $t\mapsto (R_{t/d})_{\ast }\unicode[STIX]{x1D709}_{d}$ , $t\in [0,1]$ .

The first point comes directly from the classification of isotopy classes of tight contact structures on $\mathbb{T}^{3}$ , that we now recall. The first result, proved in [Gir94], is that all incompressible pre-Lagrangian tori in $(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{d})$ are isotopic to $\{z=0\}$ . In particular, they share a common homology class which is well defined up to sign in $H_{2}(\mathbb{T}^{3})$ . Next recall that the torsion of a contact manifold $(V,\unicode[STIX]{x1D709})$ was defined, in [Gir00, Definition 1.2], to be the supremum of all integers $n\geqslant 1$ such that there exist a contact embedding of

$$\begin{eqnarray}(T^{2}\times [0,1],\ker (\cos (2n\unicode[STIX]{x1D70B}z)\,dx-\sin (2n\unicode[STIX]{x1D70B}z)\,dy)),\quad (x,y,z)\in T^{2}\times [0,1]\end{eqnarray}$$

into the interior of $(V,\unicode[STIX]{x1D709})$ , or zero if no such integer $n$ exists. It follows from [Gir00, Proposition 3.42] that the torsion of $\unicode[STIX]{x1D709}_{d}$ on $\mathbb{T}^{3}$ is $d-1$ . The classification of isotopy classes of tight contact structure on $\mathbb{T}^{3}$ established in [Gir00] is that any tight contact structure is isomorphic to some $\unicode[STIX]{x1D709}_{d}$ , and two of them are isotopic if and only if they have the same torsion and their incompressible pre-Lagrangian tori are homologous. The first point of the above theorem follows from this classification, and the obvious observation that isomorphic contact structures have the same torsion.

The description of the kernel in the second point has exactly the same proof as in the preceding section.

The third point is slightly different because $\unicode[STIX]{x1D70B}_{1}({\mathcal{D}}(\mathbb{T}^{3}),\text{Id})$ has rank three. It is generated by the three obvious circle actions on $(\mathbb{S}^{1})^{3}$ . However, two of these circle actions actually belong to ${\mathcal{D}}(\mathbb{T}^{3};\unicode[STIX]{x1D709}_{d})$ so that, in the exact sequence

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{1}({\mathcal{D}}(\mathbb{T}^{3}),\text{Id})\rightarrow \unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}},\unicode[STIX]{x1D709}_{d})\rightarrow \unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(\mathbb{T}^{3};\unicode[STIX]{x1D709}_{d}))\rightarrow \unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(\mathbb{T}^{3}))\end{eqnarray}$$

considered in the proof of Corollary 2.6, the extra generators of $\unicode[STIX]{x1D70B}_{1}({\mathcal{D}}(\mathbb{T}^{3},\text{Id}))$ are mapped to trivial elements of $\unicode[STIX]{x1D70B}_{1}({\mathcal{C}}{\mathcal{S}},\unicode[STIX]{x1D709}_{d})$ , and the end result does not change.

3 Examples of disconnected spaces of embeddings

3.1 Erratum about the reference [Gir01b]

In the proof of [Gir01b, Proposition 10], the assertion following the words Grâce au lemme 14 (namely, the claim that il est possible de trouver des points $s_{0}=0<s_{1}<\cdots <s_{k}=1$ tels que, pour $0\leqslant i\leqslant k-1$ , les surfaces $F_{s}$ , $s\in [s_{i},s_{i+1}]$ soient toutes incluses dans un voisinage rétractile $U_{i}$ de $F_{s_{i}}$ ) is wrong. In fact, the statement of Proposition 10 turns out to be wrong (Proposition 3.5 below provides a counter-example). As a consequence, Lemma 19 and the proofs of Theorems 1, 3 and 4 are also wrong. Other intermediate results (in particular, Lemma 15 which is sometimes useful as a complement to contact convexity theory) are not impacted.

To be more explicit, Lemma 14 allows us to assume that each surface $F_{s}$ has a retractible neighborhood $U_{s}$ . Each neighborhood $U_{s}$ contains all surfaces sufficiently close to $F_{s}$ ; in other words, there exists a positive number $\unicode[STIX]{x1D700}_{s}$ such that $F_{t}\subset U_{s}$ as soon as $|s-t|<\unicode[STIX]{x1D700}_{s}$ . The intervals $J_{s}=(s-\unicode[STIX]{x1D700}_{s},s+\unicode[STIX]{x1D700}_{s})$ form an open covering of the segment $[0,1]$ so this covering admits a finite subcovering $J_{s_{i}}$ , $0\leqslant i\leqslant k$ , and the points $s_{i}$ can be chosen and indexed so that $s_{0}=0$ , $s_{k}=1$ and $s_{i-1}<s_{i}$ for $1\leqslant i\leqslant k$ . Thus, each $U_{i}=U_{s_{i}}$ is a retractible neighborhood of $F_{s_{i}}$ such that $U_{i}\supset F_{s}$ for all $s$ with $|s-s_{i}|<\unicode[STIX]{x1D700}_{s_{i}}$ . This does not imply that $U_{i}\supset F_{s}$ for all $s\in [s_{i},s_{i+1}]$ and, in general, it is impossible to force $U_{i}$ to contain $F_{s_{i+1}}$ . Here is an example: assume that $F_{s}$ is a convex torus for $s\notin \{1/3,2/3\}$ and that $s=1/3$ (respectively $s=2/3$ ) corresponds to the death (respectively the birth) of a pair of parallel components in the dividing set; then $F_{1/3}$ cannot appear in any retractible neighborhood of any $F_{s}$ with $s<1/3$ . Indeed [Gir00, Lemma 4.7] guarantees, in particular, that a homogeneous neighborhood of a torus divided by $2n$ essential curve contains no essential torus divided by $2k$ curves with $k<n$ . In contrast, $F_{2/3}$ can appear in a retractible neighborhood of some $F_{s}$ with $1/3<s<2/3$ , see [Gir00, Lemma 3.31b].

3.2 Disconnected spaces of embeddings

In this section we describe examples of disconnected spaces consisting of smoothly isotopic embeddings inducing a fixed characteristic foliation. Those examples should be compared with the connectedness results which were crucial in § 2, and complement the erratum above. More specifically, we construct disconnected spaces of smoothly isotopic $\unicode[STIX]{x1D709}$ -convex embeddings with a fixed dividing set, and the former spaces are deformation retracts of the latter by Proposition 1.4.

Proposition 3.1. In $\mathbb{S}^{3}$ equipped with its standard contact structure, let $D$ be an unknotted immersed disk with a single clasp self-intersection and such that the contact structure is tangent to $D$ along its boundary (here unknotted means that $D$ has a regular neighborhood which is an unknotted solid torus, see Figures 1 and 2). Let $W$ be any unknotted solid torus which is a regular neighborhood of $D$ . Assume that $T=\unicode[STIX]{x2202}W$ is $\unicode[STIX]{x1D709}$ -convex. The space ${\mathcal{P}}_{\!\!o}(T,\mathbb{S}^{3};\unicode[STIX]{x1D709})$ is not connected.

Proof. We coorient $T$ so that $W$ is on the negative side of $T$ and we denote by $W^{\prime }$ the solid torus which is the closure of $\mathbb{S}^{3}\setminus W$ . Since $\unicode[STIX]{x1D709}$ orients $\mathbb{S}^{3}$ , we also get an orientation on $T$ . This orientation induces a cyclic ordering on $P(H_{1}(T;\mathbb{R}))$ . We set $d=\ker (H_{1}(T)\rightarrow H_{1}(W))$ and $d^{\prime }=\ker (H_{1}(T)\rightarrow H_{1}(W^{\prime }))$ where maps are induced by inclusion. A direction in $H_{1}(T)$ distinct from $d$ and $d^{\prime }$ will be called positive if it lies between $d$ and $d^{\prime }$ , and negative otherwise.

If $T_{1}$ is any cooriented unknotted torus, then we can repeat the above discussion and, for any isotopy sending $T$ to $T_{1}$ (preserving orientations), positive directions will get identified with positive directions because such isotopies map meridian disks to meridian disks in each solid torus.

Claim. Any unknotted $\unicode[STIX]{x1D709}$ -convex torus in $\mathbb{S}^{3}$ is divided by a collection of essential closed curves whose direction is positive.

Figure 1. An unknotted immersed disk with a single clasp, sitting inside an unknotted solid torus.

Figure 2. Lagrangian projection of the boundary of an immersed overtwisted disk in the standard contact $\mathbb{R}^{3}$ .

Proof of claim.

Because $\unicode[STIX]{x1D709}$ is tight, we know from [Gir01a, Théorème 4.5] that $T$ is divided by a collection of parallel homotopically essential circles. We fix a positive basis $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D707}^{\prime })$ of $H_{1}(T)$ such that $\unicode[STIX]{x1D707}$ is in $d$ and $\unicode[STIX]{x1D707}^{\prime }$ is in $d^{\prime }$ (using the notation above). Let $\unicode[STIX]{x1D6E4}$ be a dividing set for $T$ . For some choice of orientation, the components of $\unicode[STIX]{x1D6E4}$ have homology class $p\unicode[STIX]{x1D707}+q\unicode[STIX]{x1D707}^{\prime }$ with $q\geqslant 0$ . The realization lemma (recalled as part of Proposition 1.4) allows to perturb $T$ so that the characteristic foliation $\unicode[STIX]{x1D709}T$ has a circle of singularities parallel to $\unicode[STIX]{x1D6E4}$ . Such a circle $L$ is a Legendrian $(p,q)$ torus knot along which $\unicode[STIX]{x1D709}$ does not twist compared with $T$ . The Seifert framing of $L$ differs from the framing coming from $T$ by $pq$ , so that the Thurston–Bennequin invariant of $L$ is $-pq$ . Since the genus of $L$ is $(|p|-1)(q-1)/2$ , the Bennequin inequality gives $-pq\leqslant |p|q-|p|-q$ . This condition is equivalent to $p>0$ and $(q-1)(p-1)\geqslant 1/4$ and, since $p$ and $q$ are integer, it is equivalent to $p\geqslant 1$ and $q\geqslant 1$ .◻

We can see $\mathbb{S}^{3}$ as the union of two unknotted curves transverse to $\unicode[STIX]{x1D709}$ and an open interval of pre-Lagrangian tori whose directions sweep out all positive directions. For each rational positive direction $d$ and each positive integer $n$ , we can perturb the corresponding pre-Lagrangian torus to a $\unicode[STIX]{x1D709}$ -convex torus $T^{\prime }$ divided by $2n$ curves with direction $d$ . So one of those $T^{\prime }$ has the same dividing set as $T$ up to isotopy. After using once more the realization lemma, we can ensure that $T^{\prime }$ is the image of $T$ under some embedding $j\in {\mathcal{P}}_{\!\!o}(T,\mathbb{S}^{3};\unicode[STIX]{x1D709})$ . But the complement of $T^{\prime }$ is universally tight, whereas $\unicode[STIX]{x1D709}_{|W}$ becomes overtwisted in a two-fold cover. So $j$ is not in the component of the inclusion in ${\mathcal{P}}_{\!\!o}(T,\mathbb{S}^{3};\unicode[STIX]{x1D709})$ .◻

Remark 3.2. In order to get the weaker result that some solid torus $W$ satisfies the conclusion of the above proposition, it is sufficient to observe that the complement of $D$ contains an unknotted Legendrian knot $L$ entwining $D$ , and define $W$ as the complement of a standard neighborhood of $L$ . In that case, we already control the dividing set of $\unicode[STIX]{x2202}W$ by construction. Note also that the knot $L$ is not isotopic to the canonical Legendrian unknot $L_{0}$ , since the complement of the later is universally tight. The classification of Legendrian unknots in [EF98] guarantees that $L$ is stabilization of $L_{0}$ . So, in order to entwine $D$ , one needs a somewhat tortuous Legendrian unknot.

Next we want to describe examples where we have explicit smooth isotopies among surfaces which are all $\unicode[STIX]{x1D709}$ -convex except for a finite number of times, and exhibit various behaviors for those isotopies. We also want to highlight situations where persistent intersection phenomena occur, and situations where a contact isotopy exists in the ambient manifold, but not inside a smaller manifold (where a smooth isotopy still exists). For all this we need the following technical definition.

Definition 3.3. A discretized isotopy of embeddings of an oriented surface $S$ into a contact 3-manifold $(V,\unicode[STIX]{x1D709})$ is an isotopy of embeddings $j:S\times [0,1]\rightarrow V$ such that, for some (unique) integer $n$ :

1. – the restriction of $j$ to $S\times [i/n,(i+1)/n]$ is an embedding for each $i$ from $0$ to $n-1$ ;

2. – all surfaces $j_{t}(S)$ are $\unicode[STIX]{x1D709}$ -convex except when $t=i/n+1/(2n)$ for some integer $i$ between $0$ and $n-1$ .

Each embedding of $S\times [i/n,(i+1)/n]$ is called a step of the discretized isotopy. It is called a forward or backward step depending on whether it is orientation preserving or reversing.

Colin’s idea, described in our proof of Proposition 2.4, combines with [Gir01b, Lemme 15] to prove that any isotopy of embeddings which starts and ends at $\unicode[STIX]{x1D709}$ -convex embeddings is homotopic, relative to its end points, to a discretized isotopy.

Any discretized isotopy $j$ defines a sequence of isotopy classes of multi-curves $\unicode[STIX]{x1D6E4}_{0},\ldots ,\unicode[STIX]{x1D6E4}_{n}$ such that the characteristic foliation of $j_{i/n}(S)$ is divided by $j_{i/n}(\unicode[STIX]{x1D6E4}_{i})$ .

Our examples below will use facts about $\mathbb{S}^{1}$ -invariant contact structures on circle bundles, which we now recall. Let $S$ be an oriented surface with non-empty boundary, and $V=S\times \mathbb{S}^{1}$ seen as a circle bundle over $S$ . Let $\unicode[STIX]{x1D6E4}$ be a properly embedded multi-curve on $S$ such that components of the complement of $\unicode[STIX]{x1D6E4}$ are labelled by plus or minus, so that adjacent components have different signs. Lutz proved, in [Lut77], that there is a cooriented $\mathbb{S}^{1}$ -invariant positive contact structure $\unicode[STIX]{x1D709}$ on $V$ which is tangent to fibers exactly along $\unicode[STIX]{x1D6E4}\times \mathbb{S}^{1}$ , and positively (respectively negatively) transverse to fibers over positive (respectively negative) components of $S\setminus \unicode[STIX]{x1D6E4}$ . One says that $\unicode[STIX]{x1D6E4}$ is the dividing set of $\unicode[STIX]{x1D709}$ (it is indeed a dividing set for each surface $S\times \{\unicode[STIX]{x1D703}\}$ ). Lutz also proved that two invariant contact structures which agree near $\unicode[STIX]{x2202}V$ and have the same dividing set are isotopic relative to $\unicode[STIX]{x2202}V$ . Let $\unicode[STIX]{x1D709}$ be such a contact structure. One can check that, for any properly embedded curve $\unicode[STIX]{x1D6FE}$ in $S$ which intersects the dividing set $\unicode[STIX]{x1D6E4}$ transversely (along a non-empty subset), the surface $\unicode[STIX]{x1D6FE}\times \mathbb{S}^{1}$ is $\unicode[STIX]{x1D709}$ -convex and divided by $(\unicode[STIX]{x1D6FE}\cap \unicode[STIX]{x1D6E4})\times \mathbb{S}^{1}$ . Proposition 1.4 and Bennequin’s theorem in [Ben83] can be used to prove that $\unicode[STIX]{x1D709}$ is tight if and only if $\unicode[STIX]{x1D6E4}$ has no homotopically trivial component or $S$ is a disk and $\unicode[STIX]{x1D6E4}$ is connected, see [Gir01a, Proposition 4.1b]. In addition, two tight $\mathbb{S}^{1}$ -invariant contact structures on $V$ are isotopic (relative to $\unicode[STIX]{x2202}V$ ) if and only if their dividing sets are isotopic (relative to $\unicode[STIX]{x2202}S$ ). This is stated only for closed surfaces in [Gir01a, Théorème 4.4b], but the proof is only easier if the boundary of $S$ is not empty.

Recall from § 2.2 that the contact structures $\unicode[STIX]{x1D709}_{d}$ on $\mathbb{T}^{3}$ with coordinates $(x,y,z)$ are defined by

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{d}=\ker (\cos (2d\unicode[STIX]{x1D70B}z)\,dx-\sin (2d\unicode[STIX]{x1D70B}z)\,dy)\end{eqnarray}$$

and they are pairwise non-isomorphic.

Proposition 3.4. In $(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{d})$ , let $T$ be the torus $\{8d\,z=\cos x\}$ . Denote by $j_{0}$ the inclusion of $T$ into $\mathbb{T}^{3}$ and by $j_{1}$ the embedding obtained by restriction to $T$ of the rotation $(x,y,z)\mapsto (x,y,z+1/d)$ . Those two embeddings are smoothly isotopic and:

1. – $j_{0}$ and $j_{1}$ induce the same characteristic foliation on $T$ ;

2. – $j_{0}$ is not isotopic to $j_{1}$ among $\unicode[STIX]{x1D709}_{d}$ -convex embeddings;

3. – there is a discretized isotopy from $j_{0}$ to $j_{1}$ with only forward steps changing the direction of dividing curves;

4. – there is a discretized isotopy from $j_{0}$ to $j_{1}$ consisting of four forward steps which change the number of dividing curves without changing their direction.

Proof. Since the rotation map is a contactomorphism, $j_{0}$ and $j_{1}$ induce the same characteristic foliation on $T$ . Assume for contradiction that $j_{0}$ and $j_{1}$ are isotopic through $\unicode[STIX]{x1D709}_{d}$ -convex surfaces. Proposition 1.4(c) and Lemma 1.2 then imply that there is a contact isotopy $\unicode[STIX]{x1D711}$ such that $j_{1}=\unicode[STIX]{x1D711}_{1}\circ j_{0}$ . We lift this isotopy to $\mathbb{T}^{2}\times \mathbb{R}$ which covers $\mathbb{T}^{3}$ by $(x,y,s)\mapsto (x,y,s~\mod ~2\unicode[STIX]{x1D70B})$ . We denote by $\unicode[STIX]{x1D711}^{\prime }$ the lifted isotopy and by $T^{\prime }$ some (fixed) lift of $T$ . We denote by $\unicode[STIX]{x1D70F}_{n}$ the translation $(x,y,s)\mapsto (x,y,s+n)$ and by $T_{[a,b]}$ the compact manifold bounded by $\unicode[STIX]{x1D70F}_{a}(T^{\prime })$ and $\unicode[STIX]{x1D70F}_{b}(T^{\prime })$ . Because $T^{\prime }$ is compact, and contact isotopies can be cut-off, we can assume that $\unicode[STIX]{x1D711}^{\prime }$ is compactly supported. Then there is some $N$ such that $\unicode[STIX]{x1D711}_{1}$ sends $T_{[-N,0]}$ to $T_{[-N,1]}$ . In particular, those submanifolds are contactomorphic. This contradicts the classification of tight contact structures on $\mathbb{T}^{3}$ , since this contactomorphism could be used to build a contactomorphism from $(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{N+1})$ to $(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{N+2})$ .

The existence of a discretized isotopy from $j_{0}$ to $j_{1}$ consisting of forward steps changing the direction of dividing curves follows from repeated uses of a small part of the classification of tight contact structures on thickened tori: if $\unicode[STIX]{x1D709}$ is a tight contact structure on $\mathbb{T}^{2}\times [0,1]$ such that $\mathbb{T}^{2}\times \{0\}$ and $\mathbb{T}^{2}\times \{1\}$ are $\unicode[STIX]{x1D709}$ -convex with two dividing curves $\unicode[STIX]{x1D6FE}_{0},\unicode[STIX]{x1D6FE}_{0}^{\prime }$ and $\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{1}^{\prime }$ respectively, such $\unicode[STIX]{x1D6FE}_{0}$ intersects $\unicode[STIX]{x1D6FE}_{1}$ transversely at one point, then $\unicode[STIX]{x1D709}$ is isotopic, relative to the boundary, to a contact structure $\unicode[STIX]{x1D709}^{\prime }$ such that all tori $\mathbb{T}^{2}\times \{t\}$ are $\unicode[STIX]{x1D709}^{\prime }$ -convex except $\mathbb{T}^{2}\times \{1/2\}$ .

In order to construct a discretized isotopy where the direction of dividing curves is constant, we see $\unicode[STIX]{x1D709}_{d}$ as an $\mathbb{S}^{1}$ -invariant contact structure on $\mathbb{T}^{3}$ with $\mathbb{S}^{1}$ action given by rotation in the $y$ direction. In order to describe an $\mathbb{S}^{1}$ -equivariant isotopy of embeddings of $T$ , it is enough to give an isotopy of curves in $\mathbb{T}^{2}$ . Curves corresponding to $\unicode[STIX]{x1D709}_{d}$ -convex tori are exactly those which are transverse to $\unicode[STIX]{x1D6E4}=\{x\in (\unicode[STIX]{x1D70B}/d)\mathbb{Z}\}$ . Figure 3 then finishes the proof.◻

Figure 3. Discretized isotopy of curves lifting to tori in $(\mathbb{T}^{3},\unicode[STIX]{x1D709}_{d})$ . Curves lifting to non-convex tori are dashed.

In our next example, the discretized isotopy oscillates, and there is persistent intersection.

Proposition 3.5. Let $V$ be the torus bundle over $\mathbb{S}^{1}$ with monodromy $B=\left(\!\begin{smallmatrix}5 & 1\\ 4 & 1\end{smallmatrix}\!\right)$ , i.e.

$$\begin{eqnarray}V=(\mathbb{T}^{2}\times \mathbb{R})/((Bx,t)\sim (x,t+1)).\end{eqnarray}$$

Let $T$ be the image of $\mathbb{T}^{2}\times \{1/2\}$ in $V$ , and let $j_{0}$ be the inclusion map from $T$ to $V$ . There is a tight virtually overtwisted contact structure $\unicode[STIX]{x1D709}$ on $V$ , and an embedding $j_{1}\in {\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ such that:

1. – $j_{0}$ is not isotopic to $j_{1}$ in ${\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ ;

2. – any $j\in {\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ such that $j(T)$ is disjoint from $T$ is isotopic to $j_{0}$ in ${\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ (in particular, $j_{1}(T)$ cannot be disjoined from $T$ by contact isotopy);

3. – there is a discretized isotopy from $j_{0}$ to $j_{1}$ with one forward step and one backward step, both modifying the direction of dividing curves.

Proof. We will use the theory of normal forms for tight contact structures on $V$ established in [Gir00, § 3] (see also [Hon00]). With the notation used in [Gir00], $V=T_{A}^{3}$ where $A=B^{-1}=\left(\!\begin{smallmatrix}1 & -1\\ -4 & 5\end{smallmatrix}\!\right)$ . Other choices of monodromies are possible, we only want to explain one simple example.

We will describe the lifts of relevant contact structures on $\mathbb{T}^{2}\times [0,1]\subset \mathbb{T}^{2}\times \mathbb{R}$ . In [Gir00], tight contact structures on thickened tori are described using two types of building blocks: rotation sequences and orbit flips, that we will briefly review.

Recall that a foliation $\unicode[STIX]{x1D70E}$ on a torus $T$ is called a suspension if there is a circle which transversely intersects all leaves. It then has an asymptotic direction $d(\unicode[STIX]{x1D70E})$ , which is a line through the origin in $H_{1}(T;\mathbb{R})$ spanned by limits of renormalized very long orbits of a directing vector field.

We fix a contact structure $\unicode[STIX]{x1D709}$ on $T\times [0,1]$ , and set $T_{a}=T\times \{a\}$ . An interval $J\subset [0,1]$ is called a rotation sequence for $\unicode[STIX]{x1D709}$ if all characteristic foliations $\unicode[STIX]{x1D709}T_{t}$ , $t\in J$ are suspensions. We say that $J$ is minimally twisting if the directions $d(\unicode[STIX]{x1D709}T_{t})$ do not sweep out the full projective line $P(H_{1}(T;\mathbb{R}))$ . Theorem 3.3 from [Gir00] guarantees that two contact structures on $T\times J$ which agree along the boundary and have $J$ as a minimally twisting rotation sequence are isotopic on $T\times J$ relative to boundary.

An interval $[a,b]\subset [0,1]$ is an orbit flip sequence for $\unicode[STIX]{x1D709}$ , with homology class $d\in H_{1}(T;\mathbb{Z})$ , if:

1. – $\unicode[STIX]{x1D709}T_{a}$ is a Morse–Smale suspension with two closed orbits whose homology classe is $d$ ;

2. – $\unicode[STIX]{x1D709}T_{b}$ is a Morse–Smale suspension with two closed orbits whose homology classe is $-d$ ;

3. – there is a multi-curve which divides all $\unicode[STIX]{x1D709}T_{t}$ , $t\in J$ .

The uniqueness lemma [Gir00, Lemma 2.7] ensures that two contact structures on $T\times [a,b]$ which agree along the boundary, and admit $[a,b]$ as an orbit flip sequence, are isotopic relative to boundary. There is an explicit model in [Gir00, § 1.F] where all $\unicode[STIX]{x1D709}T_{t}$ are suspensions except one which has two circles of singularities instead of regular closed leaves.

Here we need two (isotopic) contact structures on $\mathbb{T}^{2}\times [0,1]$ . We fix a Morse–Smale suspension $\unicode[STIX]{x1D70E}_{0}$ on $\mathbb{T}^{2}$ with two closed orbits having homology class $(1,0)$ , and we denote by $\unicode[STIX]{x1D70E}_{1}$ the image of $\unicode[STIX]{x1D70E}_{0}$ under $A$ . We also fix a Morse–Smale suspension $\unicode[STIX]{x1D70E}_{1/2}$ with two closed orbits having homology class $(-1,1)$ . Let $\unicode[STIX]{x1D709}$ be a contact structure on $\mathbb{T}^{2}\times [0,1]$ such that:

1. – $\unicode[STIX]{x1D709}$ prints $\unicode[STIX]{x1D70E}_{t}$ on $\mathbb{T}^{2}\times \{t\}$ for $t\in \{0,1/2,1\}$ ;

2. – $[0,1]$ is a union of minimally twisting rotation sequences and two orbit flip sequences with homology classes $(1,0)$ and $(-1,1)$ respectively.

Let $\unicode[STIX]{x1D709}^{\prime }$ be a contact structure with the same properties except that orbit flip homology classes are $(1,-1)$ and $(-1,2)$ . The explicit construction of [Gir00, Example 3.41] guarantees that $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D709}^{\prime }$ are isotopic (relative to the boundary). More specifically, it builds a contact structure printing a non-generic movie of characteristic foliations where two saddle connections happen on the same torus, and such that the movies printed by $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D709}^{\prime }$ are essentially obtained by choosing the order in which these connections appear. Alternatively, one can see the isotopy between $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D709}^{\prime }$ as an application of the ‘shuffling lemma’ of Honda [Hon00, Lemma 4.14]. We will also denote by $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D709}^{\prime }$ the induced contact structures on $V$ . And we denote by $T$ the image in $V$ of $\mathbb{T}_{1/2}^{2}$ .

Let $\unicode[STIX]{x1D711}$ be a smooth isotopy of $V$ such that $\unicode[STIX]{x1D709}^{\prime }=\unicode[STIX]{x1D711}_{1}^{\ast }\unicode[STIX]{x1D709}$ . Assume for contradiction that $j_{0}:T{\hookrightarrow}V$ and $j_{1}=\unicode[STIX]{x1D711}_{1}\circ j_{0}$ are in the same component of ${\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ . Using the path lifting property for the map ${\mathcal{D}}_{\!o}(V;\unicode[STIX]{x1D709})\rightarrow {\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ guaranteed by Lemma 1.2, we get a contact isotopy $\unicode[STIX]{x1D703}$ for $\unicode[STIX]{x1D709}$ such that $j_{1}=\unicode[STIX]{x1D703}_{1}\circ j_{0}$ . Then $\unicode[STIX]{x1D713}:=\unicode[STIX]{x1D703}_{1}^{-1}\circ \unicode[STIX]{x1D711}_{1}$ is a diffeomorphism relative to $T$ , and pulls back $\unicode[STIX]{x1D709}$ to $\unicode[STIX]{x1D709}^{\prime }$ . Thus, we can cut $V$ along $T$ to get a thickened torus $Y$ , naturally identified with $\mathbb{T}^{2}\times [1/2,3/2]$ . The diffeomorphism $\unicode[STIX]{x1D713}$ induces a diffeomorphism of $Y$ which is relative to the boundary, hence acts trivially on $H_{1}(Y)$ . This is a contradiction because the restriction of $\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D709}^{\prime }$ to $Y$ do not have the same relative Euler class in $H_{1}(Y)$ . Recall that $e(Y;\unicode[STIX]{x1D709})$ is the homology class of the vanishing locus of any generic section of $\unicode[STIX]{x1D709}$ which spans $\unicode[STIX]{x1D709}\unicode[STIX]{x2202}Y$ (with the correct orientation) along $\unicode[STIX]{x2202}Y$ . Here, contributions to this class come from orbit flips and we get $e(Y;\unicode[STIX]{x1D709})=2(-1,1)+2A(1,0)=2(0,-3)$ while $e(Y;\unicode[STIX]{x1D709}^{\prime })=2(-1,2)+2A(1,-1)=2(1,-7)$ . Note, for sanity check, that those two classes become the same in $V$ , since $e(Y;\unicode[STIX]{x1D709}^{\prime })-e(Y;\unicode[STIX]{x1D709})=2(1,-4)=(\text{Id}-A)(0,1)$ .

The second point of the proposition follows again from classification results. Let $j$ be an embedding in ${\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ such that $j(T)$ is disjoint from $T$ . The classification of incompressible tori in the complement of $T$ guarantees that $T$ and $j(T)$ bound a thickened torus $N$ in $V$ . The classification of tight contact structures on thickened tori in [Gir00] or [Hon00] ensures that either $j$ is isotopic to $j_{0}$ in ${\mathcal{P}}_{\!\!o}(T,V;\unicode[STIX]{x1D709})$ , or there exists a contact embedding of

$$\begin{eqnarray}(T^{2}\times [0,1],\ker (\cos (\unicode[STIX]{x1D70B}z)\,dx-\sin (\unicode[STIX]{x1D70B}z)\,dy)),\quad (x,y,z)\in T^{2}\times [0,1]\end{eqnarray}$$

into the interior of $N$ . But the existence of such an embedding is ruled out by the study of tight contact structure on $V$ , specifically [Gir00, Proposition 1.8].

The announced discretized isotopy uses the image of $\mathbb{T}^{2}\times \{0\}$ as a intermediate surface, and its existence is guaranteed by the classification result quoted in the proof of Proposition 3.4.◻

Finally we describe an example on a manifold with boundary with the same situation as above, but things untangle inside a larger manifold.

Proposition 3.6. Let $V$ denote the manifold $\mathbb{T}^{2}\times [0,1]$ , and $V^{\prime }=\mathbb{T}^{2}\times [0,1/2]$ . There is a universally tight contact structure $\unicode[STIX]{x1D709}$ on $V$ , and two smoothly isotopic $\unicode[STIX]{x1D709}$ -convex embeddings $j_{0},j_{1}:\mathbb{T}^{2}\rightarrow V^{\prime }$ with images $T_{0}$ and $T_{1}$ , such that:

1. – $j_{0}$ is isotopic to $j_{1}$ among $\unicode[STIX]{x1D709}$ -convex embeddings in $V$ ;

2. – $j_{0}$ is not isotopic to $j_{1}$ among $\unicode[STIX]{x1D709}$ -convex embeddings in $V^{\prime }$ ;

3. – $T_{0}$ cannot be disjoined from $T_{1}$ by an isotopy among $\unicode[STIX]{x1D709}$ -convex surfaces in $V^{\prime }$ ;

4. – there is a discretized isotopy from $j_{0}$ to $j_{1}$ in $V^{\prime }$ with one forward step and one backward step, both modifying the direction of dividing curves;

5. – there is a discretized isotopy from $j_{0}$ to $j_{1}$ in $V^{\prime }$ with one forward step and one backward step, both modifying the number of dividing curves.

Proof. The construction is pictured in Figure 4. Let $S$ be the annulus $\{1\leqslant |z|\leqslant 3\}\subset \mathbb{C}$ and $S^{\prime }\subset S$ the subannulus $\{1\leqslant |z|\leqslant 2\}$ . We fix an identification between $V$ and $S\times \mathbb{S}^{1}$ which identifies $V^{\prime }$ with $S^{\prime }\times \mathbb{S}^{1}$ . Let $\unicode[STIX]{x1D6E4}^{\prime }=\unicode[STIX]{x1D6E4}_{1}^{\prime }\cup \unicode[STIX]{x1D6E4}_{2}^{\prime }$ be a disjoint union of two properly embedded arcs in $S^{\prime }$ whose end points are on the circle $\{|z|=2\}$ . Let $\unicode[STIX]{x1D6E4}$ be a smooth homotopically essential circle in $S$ such that $\unicode[STIX]{x1D6E4}\cap S^{\prime }=\unicode[STIX]{x1D6E4}^{\prime }$ . Let $\unicode[STIX]{x1D709}$ be an $\mathbb{S}^{1}$ -invariant contact structure on $V$ with dividing set $\unicode[STIX]{x1D6E4}$ , and denote by $\unicode[STIX]{x1D709}^{\prime }$ its restriction to $V^{\prime }$ . Let $\unicode[STIX]{x1D6FE}_{0}$ and $\unicode[STIX]{x1D6FE}_{1}$ be homotopically essential circles in $S^{\prime }$ such that $\unicode[STIX]{x1D6FE}_{i}$ intersects transversely $\unicode[STIX]{x1D6E4}_{i}^{\prime }$ in two points, and does not intersect the other component of $\unicode[STIX]{x1D6E4}^{\prime }$ . The tori we want are $T_{0}=\unicode[STIX]{x1D6FE}_{0}\times \mathbb{S}^{1}$ and $T_{1}=\unicode[STIX]{x1D6FE}_{1}\times \mathbb{S}^{1}$ , parametrized by product maps.

Figure 4. The example of Proposition 3.6. The dividing set $\unicode[STIX]{x1D6E4}$ is the thick curve, $\unicode[STIX]{x1D6FE}_{0}$ and $\unicode[STIX]{x1D6FE}_{1}$ are dashed.

There is an isotopy through $\unicode[STIX]{x1D709}$ -convex surfaces in $V$ because $\unicode[STIX]{x1D6FE}_{0}$ and $\unicode[STIX]{x1D6FE}_{1}$ are isotopic in $S$ through curves transverse to $\unicode[STIX]{x1D6E4}$ .

Assume for contradiction that there is such an isotopy in $V^{\prime }$ . We can arrange $\unicode[STIX]{x1D709}$ so that $j_{0}$ and $j_{1}$ induce the same characteristic foliation on $\mathbb{T}^{2}$ and, using Proposition 1.4(c) and Lemma 1.2, our isotopy through convex surfaces can then be converted into a contact isotopy $\unicode[STIX]{x1D711}$ relative to the boundary. We denote by $\unicode[STIX]{x2202}_{1}V^{\prime }$ and $\unicode[STIX]{x2202}_{2}V^{\prime }$ the connected components $\{|z|=1\}\times \mathbb{S}^{1}$ and $\{|z|=2\}\times \mathbb{S}^{1}$ of $\unicode[STIX]{x2202}V^{\prime }$ . Let $\unicode[STIX]{x1D713}_{0}$ and $\unicode[STIX]{x1D713}_{1}$ be smooth embeddings of $V^{\prime }$ into itself such that:

1. – each $\unicode[STIX]{x1D713}_{i}$ is $\mathbb{S}^{1}$ -equivariant;

2. – each $\unicode[STIX]{x1D713}_{i}$ is the identity on $\unicode[STIX]{x2202}_{2}V^{\prime }$ ;

3. – $\unicode[STIX]{x1D713}_{i}(\unicode[STIX]{x2202}_{1}V^{\prime })=T_{i}$ .

The contact structures $\unicode[STIX]{x1D713}_{0}^{\ast }\unicode[STIX]{x1D709}$ and $\unicode[STIX]{x1D713}_{1}^{\ast }\unicode[STIX]{x1D709}$ on $V^{\prime }$ are $\mathbb{S}^{1}$ -invariant and the associated dividing sets $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D713}_{0}}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D713}_{1}}$ are shown on Figure 5. The contactomorphism $\unicode[STIX]{x1D711}_{1}$ then induces a contactomorphism between $(V^{\prime },\unicode[STIX]{x1D713}_{0}^{\ast }\unicode[STIX]{x1D709})$ and $(V^{\prime },\unicode[STIX]{x1D713}_{1}^{\ast }\unicode[STIX]{x1D709})$ which is the identity on $\unicode[STIX]{x2202}_{2}V^{\prime }$ . However, the classification of $\mathbb{S}^{1}$ -invariant contact structures forbids the existence of this contactomorphism. In this case, we can argue directly as follows. We denote by $S^{\prime \prime }$ the annulus $\{2\leqslant |z|\leqslant 3\}$ . Let $\unicode[STIX]{x1D709}^{\prime \prime }$ be a contact structure on $S^{\prime \prime }\times \mathbb{S}^{1}$ which is $\mathbb{S}^{1}$ -invariant and tangent to $\mathbb{S}^{1}$ along some $\unicode[STIX]{x1D6E4}^{\prime \prime }$ such that $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D713}_{0}}\cup \unicode[STIX]{x1D6E4}^{\prime \prime }$ has a homotopically trivial component but $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D713}_{1}}\cup \unicode[STIX]{x1D6E4}^{\prime \prime }$ has not. The contact structure $\unicode[STIX]{x1D709}^{\prime \prime }\cup \unicode[STIX]{x1D713}_{0}^{\ast }\unicode[STIX]{x1D709}$ on $V$ is overtwisted, whereas $\unicode[STIX]{x1D709}^{\prime \prime }\cup \unicode[STIX]{x1D713}_{1}^{\ast }\unicode[STIX]{x1D709}$ is tight, so we have a contradiction.

Figure 5. Dividing curves for the proof of Proposition 3.6. Thin curves are boundary components of $S^{\prime }$ and thick curves are the components of the dividing sets.

So there is no contact isotopy $\unicode[STIX]{x1D711}$ in $V^{\prime }$ such that $j_{1}=\unicode[STIX]{x1D711}_{1}\circ j_{0}$ . Assume for contradiction that there is a contact isotopy $\unicode[STIX]{x1D711}$ in $V^{\prime }$ such that $T_{0}^{\prime }=\unicode[STIX]{x1D711}_{1}(T_{0})$ is disjoint from $T_{1}$ . The classification of incompressible surfaces in thickened tori ensures that $T_{0}^{\prime }\cup T_{1}$ is the boundary of a thickened torus in the interior of $V^{\prime }$ . After some smooth deformation, we can assume that $T_{0}^{\prime }=\{|z|=r_{0}\}\times \mathbb{S}^{1}$ and $T_{1}=\{|z|=r_{1}\}\times \mathbb{S}^{1}$ and the contact structure is $\mathbb{S}^{1}$ -invariant near $T_{0}^{\prime }$ and $T_{1}$ (note that we do not know the sign of $r_{0}-r_{1}$ ). Those tori are both divided by vertical curves $\{\ast \}\times \mathbb{S}^{1}$ , so the classification of universally tight contact structures on thickened tori guarantees that, after some further isotopy relative to $T_{0}^{\prime }$ and $T_{1}$ , the contact structure is $\mathbb{S}^{1}$ -invariant everywhere (see [Gir00, Théorème 4.4] or [Hon00]). The dividing set in the annulus $\{|z|\in [r_{0},r_{1}]\}$ intersects each boundary component in two points, so it is either two boundary parallel arcs and some closed components, or two traversing arcs. The first possibility is ruled out by the classification of $\mathbb{S}^{1}$ -invariant contact structures up to (non-necessarily invariant) isotopy since the full dividing set on $S^{\prime }$ would not be isotopic to $\unicode[STIX]{x1D6E4}^{\prime }$ . The second possibility is ruled out because $T_{0}^{\prime }$ and $T_{1}$ would then be isotopic among $\unicode[STIX]{x1D709}^{\prime }$ -convex surfaces, contradicting the previous point.

The construction of discretized isotopies is completely analogous to what we discussed for Proposition 3.4. ◻