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 [Reference LibermannLib59], 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 ${\mathcal{P}}_{\!\!o}(T,V_{d})$ .◻

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 [Reference ColinCol97], 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 [Reference GirouxGir00, Reference HondaHon00] 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 [Reference PickPic99], 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 [Reference GhigginiGhi05, Proposition 5.4] and further developed in [Reference MassotMas08, 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 [Reference MassotMas08], 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.◻

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 [Reference GirouxGir01b, 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 [Reference ColinCol99, Theorem 3.1] then replaces Proposition 2.4, and the final isotopy is provided by Eliashberg’s result in [Reference EliashbergEli92] 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.◻

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 [Reference LaudenbachLau74, Reference HatcherHat76] 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 $\unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(V_{d};\unicode[STIX]{x1D709}_{d}))\rightarrow \unicode[STIX]{x1D70B}_{0}({\mathcal{D}}(V_{d}))$ . ◻

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.◻

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 [Reference WaldhausenWal67, 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.◻

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.

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 $\mathbb{R}^{3}$ .

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 ${\mathcal{P}}_{\!\!o}(T,\mathbb{S}^{3};\unicode[STIX]{x1D709})$ .◻

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.◻

Proof. We will use the theory of normal forms for tight contact structures on $V$ established in [Reference GirouxGir00, § 3] (see also [Reference HondaHon00]). With the notation used in [Reference GirouxGir00], $V=T_{A}^{3}$ 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 [Reference GirouxGir00], 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 [Reference GirouxGir00] 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 [Reference GirouxGir00, 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 [Reference GirouxGir00, § 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 [Reference GirouxGir00, 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 [Reference HondaHon00, 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 [Reference GirouxGir00] or [Reference HondaHon00] 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 [Reference GirouxGir00, 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.◻

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 [Reference GirouxGir00, Théorème 4.4] or [Reference HondaHon00]). 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. ◻