Bordered Floer homology and contact structures

2.2.1 Earlier Heegaard Floer theoretic contact invariants

Using open books, Ozsváth and Szabó defined a Heegaard Floer invariant of a closed contact three-manifold [Reference Ozsváth and SzabóOS05]. For a contact manifold $(M,\xi )$ , this invariant is a class $c(\xi )$ in the Heegaard Floer homology $\widehat {\mathit {HF}}(-M)$ . In [Reference Honda, Kazez and MatićHKM09b], Honda, Kazez and Matić gave an alternative description of $c(\xi )$ . Their construction again uses open books and goes roughly as follows. An open book $(S,h)$ for $(M,\xi )$ induces a Heegaard splitting of M into and with Heegaard surface $\Sigma =(S\times \{1/2\})\cup _B-(S\times \{0\})$ . Let $\{a_i\}$ be a collection of properly embedded arcs on S that cut S into a disk. For all i, let $b_i$ be a small perturbation of $a_i$ that moves the endpoints in the positive direction along $\partial S$ so that $b_i$ intersects $a_i$ in one point. Fix a basepoint z on $\partial S$ away from the thin strips cobounded by the $\{a_i\}$ and $\{b_i\}$ . It is clear that $\Sigma = \partial U_1 = -\partial U_2$ and that the $\alpha _i := \partial (a_i\times \left [0,1/2\right ])$ bound compressing disks for $U_1$ and the $\beta _i := \partial (b_i\times \left [1/2,1\right ])$ bound compressing disks for $U_2$ . Thus, $(\Sigma , \boldsymbol {\alpha }, \boldsymbol {\beta }, z)$ is a Heegaard diagram for M. One can see the curves as

$$ \begin{align*} \alpha_i &= a_i\cup -a_i \subset (S\times \{1/2\})\cup_B-(S\times\{0\}),\\ \beta_i &= b_i\cup -h(b_i) \subset (S\times \{1/2\})\cup_B-(S\times\{0\}). \end{align*} $$

Then $c(\xi )\in \widehat {\mathit {HF}}(-M)$ is defined as the homotopy equivalence class of the unique generator of $\widehat {\mathit {CF}}(\Sigma , {\boldsymbol {\beta }}, {\boldsymbol {\alpha }}, z)$ fully supported on $S\times \{1/2\}$ .

Using partial open books, Honda, Kazez and Matić then extended the above construction to define an invariant $EH(M, \Gamma , \xi )$ of contact three-manifolds with convex boundary. In this construction, one begins with a partial open book $(S,P, h)$ for $(M, \Gamma , \xi )$ , where S is built up from $\overline {S\setminus P}$ by the addition of one-handles $P_i$ , as in Definition 2.1. The roles of $U_1$ and $U_2$ are now played by $\big (P\times [-1/2,0]\big )\cup \big (S\times [0,1/2]\big )/\sim $ and $\big (S\times [1/2,1]\big )\cup \big (P\times [-1,-1/2]\big )/\sim $ , respectively, and the curves $\{a_i\}$ are the cocores of the one-handles $P_i$ . The rest of the construction is as above, except that the final generator supported on $P\times \{-1/2\}$ defines an element in the sutured Floer homology $\mathit {SFH}(-M, -\Gamma )$ . Under certain technical conditions, the authors also defined a gluing map for sutured Floer homology that respects the contact invariants [Reference Honda, Kazez and MatićHKM08].

Before briefly reviewing bordered sutured Floer homology, we discuss a straightforward generalization of the contact invariant to multipointed Heegaard diagrams. We begin by constructing a Heegaard diagram analogous to the one above, except that we allow additional arcs $\{a_i\}$ that cut S into n disks. We place a basepoint in each of the disks thus obtained. This results in a multipointed Heegaard diagram $(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathbf {z})$ for M. The unique generator $\mathbf {x}$ on $S\times \{1/2\}$ defines an element in ${\mathrm {\widetilde {HF}}}(\Sigma , {\boldsymbol {\beta }}, {\boldsymbol {\alpha }}, \mathbf {z}) \cong \widehat {\mathit {HF}}(-M)\otimes H_*(T^{n-1})$ . Here, ${\mathrm {\widetilde {HF}}}$ is the homology with respect to the differential that avoids all basepoints, and $H_*(T^{n-1}) \cong H_*(S^1)^{\otimes (n-1)}$ is the ordinary singular homology of $T^{n-1}$ .

An adaptation of the argument of part (5) of [Reference Baldwin, Vela-Vick and VértesiBVVV13, Theorem 3.1] shows the following.

Proposition 2.3. Let $(S,h)$ and $(S',h')$ be two open book decompositions compatible with $(M,\xi )$ . Let $\{a_i\}$ and $\{a_i'\}$ be sets of cutting arcs that cut up S and $S'$ into n and $n'$ disks, respectively, with $n<n'$ . The graded isomorphism between the Heegaard Floer homologies induced by Heegaard moves, including index $0$ and $3$ stabilizations,

$$\begin{align*}{\mathrm {\widetilde{HF}}}(\Sigma', {\boldsymbol{\beta}}', {\boldsymbol{\alpha}}', \mathbf{z}') \otimes H_*(T^{n'-n}) \to {\mathrm {\widetilde{HF}}}(\Sigma, {\boldsymbol{\beta}}, {\boldsymbol{\alpha}}, \mathbf{z}) \end{align*}$$

maps the homology class to $[\mathbf {x}'] \otimes \theta ^{\otimes {(n'-n)}}$ to $[\mathbf {x}]$ , where $\theta $ corresponds to the lower-degree generator of $H_*(S^1)$ .

This means that up to homotopy equivalence, the multipointed contact invariant is simply $c(\xi )\otimes \theta ^{\otimes {(n-1)}}$ , where n is the number of basepoints.

We will also make use of the fact that the multipointed Heegaard Floer homology for a closed three-manifold M can be interpreted as the sutured Floer homology of M with balls removed, as follows. Let $(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathbf {z})$ be a multipointed Heegaard diagram for M with n basepoints, and for each basepoint $z\in \mathbf {z}$ , let $D^2_z$ be a disk neighborhood of z. Then $(\Sigma \setminus \cup _{\mathbf {z}} D^2_z, {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ is a sutured Heegaard diagram for $M(n) = (M\setminus \cup _{z\in \mathbf {z}}B^3_{z}, \cup _{z\in \mathbf {z}}\partial D^2_z)$ . As the two Heegaard diagrams are identical outside the basepointed/sutured regions, the chain complex ${\mathrm {\widetilde {CF}}}(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathbf {z})$ is isomorphic to the chain complex ${\mathrm {SFC}}(\Sigma \setminus \cup _{\mathbf {z}} D^2_z, {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ . Thus, we can compute the multipointed Heegaard Floer homology ${\mathrm {\widetilde {HF}}}(\Sigma , {\boldsymbol {\beta }}, {\boldsymbol {\alpha }}, \mathbf {z})$ as the sutured Floer homology $\mathit {SFH}(\Sigma \setminus \cup _{\mathbf {z}} D^2_z, {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ . See also [Reference JuhászJuh06, Proposition 9.14].

2.2.2 Bordered sutured Floer homology

Lipshitz, Oszváth and Thurston refine Heegaard Floer homology to a bordered variant associated to a three-manifold with parametrized boundary [Reference Lipshitz, Ozsvath and ThurstonLOT18, Reference Lipshitz, Ozsváth and ThurstonLOT11], and Zarev [Reference ZarevZar09, Reference ZarevZar10] further refines the invariant to an invariant of sutured manifolds with partially parameterized boundary. We briefly discuss Zarev’s constructions [Reference ZarevZar10, Section 3]. Note that the following description is indicative, rather than complete; for a review of the algebraic definitions involved, see [Reference Lipshitz, Ozsvath and ThurstonLOT18, Section 2].

Recall that an arc diagram $\mathcal Z = (Z,a,m)$ consists of a finite collection of oriented line segments Z, commonly called arcs in the literature, a collection of points $a=(a_1,\cdots , a_{2k})$ on Z, a matching m of the points in a into pairs and a ‘type,’ $\alpha $ or $\beta $ . To every arc diagram $\mathcal Z$ one associates an $\mathcal A_{\infty }$ -algebra $\mathcal A(\mathcal Z)$ generated by tuples of oriented arcs in $[0,1]\times Z$ such that each arc connects some $(0,a_i)$ to some $(1, a_j)$ with $a_j\geq a_i$ , up to an equivalence relation imposed by the matching. The ground ring of idempotents $\mathcal I(\mathcal Z)$ of this algebra consists of elements $\iota $ corresponding to tuples of horizontal strands $[0,1]\times \{a_k\}$ in $[0,1]\times Z$ . One further associates to $\mathcal Z$ a graph $G(\mathcal Z)$ and a surface $F(\mathcal Z)$ . The graph is constructed by beginning with Z and adding an edge between each pair of matched points, while $F(\mathcal Z)$ is constructed from $Z \times \left [0,1\right ]$ by attaching one-handles to neighborhoods of the matched points on $Z\times \{0\}$ .

A bordered sutured manifold $(M,\mathsf {\Gamma },\mathcal Z)$ consists of the following:

• • a three-manifold M whose boundary decomposes as a bordered part F and a sutured part T;

• • an arc diagram $\mathcal {Z}=(Z,a,m)$ and an identification of F with $F(\mathcal Z)$ ;

• • a dividing set $\mathsf {\Gamma }$ which is a properly embedded, oriented one-manifold in T with boundary $\partial \mathsf {\Gamma } = -\partial (Z\times \{\frac 1 2 \})$ so that $\mathsf {\Gamma }$ decomposes T into the union of $R_+(\mathsf {\Gamma })$ and $R_-(\mathsf {\Gamma })$ with $\partial R_{\pm }(\mathsf {\Gamma })\setminus \partial F=\pm \mathsf {\Gamma }$ .

The components of the dividing set are also referred to as sutures.

A $\beta $ -bordered Heegaard diagram $\mathcal {H}=(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathcal Z)$ consists of a compact surface $\Sigma $ with no closed components; a collection of pairwise disjoint, properly embedded circles ${\boldsymbol {\alpha }}$ ; a collection of pairwise disjoint, properly embedded circles ${\boldsymbol {\beta }}^{c}$ and of properly embedded arcs ${\boldsymbol {\beta }}^{a}$ , with ${\boldsymbol {\beta }} = {\boldsymbol {\beta }}^a \cup {\boldsymbol {\beta }}^c$ and an arc diagram $\mathcal Z$ of $\beta $ -type, together with an embedding $G(\mathcal Z)\rightarrow \Sigma $ which maps Z to a subset of $\partial \Sigma $ and the edges of $G(\mathcal Z)$ connecting matched points to the arcs ${\boldsymbol {\beta }}^c$ . One also requires that the maps $\pi _0(\partial \Sigma \setminus Z)\to \pi _0(\Sigma \setminus {\boldsymbol {\alpha }})$ and $\pi _0(\partial \Sigma \setminus Z)\to \pi _0(\Sigma \setminus {\boldsymbol {\beta }})$ be surjective. (An $\alpha $ -bordered diagram is defined similarly, mutatis mutandis.) One constructs a bordered sutured manifold $(M,\mathsf {\Gamma },\mathcal Z)$ from $\mathcal {H}$ as follows. The three-manifold M is obtained by attaching two-handles to $\Sigma \times [0,1]$ along ${\boldsymbol {\alpha }} \times \{1\}$ and ${\boldsymbol {\beta }}^c\times \{0\}$ circles; the dividing set $\mathsf {\Gamma }$ appears as $\mathsf {\Gamma } = (\partial \Sigma \setminus Z) \times \left \{\frac {1}{2}\right \}$ , and $F(\mathcal Z)$ is a neighborhood of $(Z \times [0,1])\cup ({\boldsymbol {\beta }}^a\times \{0\})$ . Another way to view $(M,\mathsf {\Gamma },\mathcal Z)$ as coming from $\mathcal {H}$ , which fits better with Morse theory, is to also attach ‘halves of two-handles’ along ${\boldsymbol {\beta }}^a \times \{0\}$ and see $F(\mathcal Z)$ as $Z\times [0,1]$ together with the intersection of the thickened cores of the partial two-handles with $\partial M$ .

Let $\textbf {x}$ and $\textbf {y}$ denote tuples of intersection points between the ${\boldsymbol {\alpha }}$ and ${\boldsymbol {\beta }}$ such that each $\alpha $ -circle is used exactly once, each $\beta $ -circle is used exactly once, and each $\beta $ -arc is used no more than once; these will ultimately be the generators of the bordered sutured modules. We recall that the set of homology classes $\pi _2(\textbf {x}, \textbf {y})$ connecting $\textbf {x}$ to $\textbf {y}$ is defined as follows. We let $I_s =[0,1]$ and $I_t = [-\infty , \infty ]$ be intervals and consider the relative homology group

$$\begin{align*}H_2(\Sigma \times I_s \times I_t, (({\boldsymbol{\alpha}} \times \{1\}) \cup ({\boldsymbol{\beta}} \times \{0\}) \cup (Z \times I_s))\times I_t), (\textbf{x}\times I_s\times \{-\infty\}) \cup (\textbf{y} \times I_s \times \{\infty\})).\end{align*}$$

The set $\pi _2(\textbf {x}, \textbf {y})$ denotes elements of this group which are sent to the fundamental class of $(\textbf {x}\times I_s)\cup (\textbf {y}\times I_s)$ by the map which applies the boundary homomorphism and then collapses the remainder of the boundary. Any homology class has a unique corresponding domain, that is, a linear combination of the components of $\Sigma \setminus ({\boldsymbol {\alpha }} \cup {\boldsymbol {\beta }})$ obtained by projection. A domain is provincial if its boundary has no intersection with Z.

The set $\pi _2(\textbf {x}, \textbf {x})$ is the set of periodic domains. We recall that $\mathcal {H}$ is said to be provincially admissible if every provincial periodic domain has both positive and negative coefficients and admissible if every periodic domain has both positive and negative coefficients.

To a provincially admissible $\beta $ -bordered Heegaard diagram $\mathcal {H}=(\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }}, \mathcal Z)$ equipped with an admissible almost complex structure, Zarev associates a left type D module $\widehat {\mathit {BSD}}(\mathcal {H})$ over $\mathcal A(-\mathcal Z)$ and right type A module $\widehat {\mathit {BSA}}(\mathcal {H})$ over $\mathcal A(\mathcal Z)$ . In both cases, the generators are tuples of intersection points as described above.Footnote ¹ The type A chain homotopy equivalence class of $\widehat {\mathit {BSA}}(\mathcal {H})$ is an invariant of the associated bordered sutured manifold $(M, \mathsf {\Gamma }, \mathcal Z)$ , written $\widehat {\mathit {BSA}}(M, \mathsf {\Gamma }, \mathcal Z)$ . More precisely, given bordered sutured Heegaard diagrams $\mathcal {H}$ and $\mathcal {H}'$ for $(M, \mathsf {\Gamma }, \mathcal Z)$ , there is a sequence of Heegaard moves connecting them as in [Reference ZarevZar09, Proposition 4.5], which induce a type A homotopy equivalence f from $\widehat {\mathit {BSA}}(\mathcal {H})$ to $\widehat {\mathit {BSA}}(\mathcal {H}')$ . This type A homotopy equivalence is a collection of maps

$$\begin{align*}f_i:\widehat{\mathit{BSA}}(\mathcal{H})\otimes \mathcal A(\mathcal Z)^{\otimes(i-1)}\to \widehat{\mathit{BSA}}(\mathcal{H}')\end{align*}$$

indexed by $i\ge 1$ , satisfying certain conditions (see [Reference Lipshitz, Ozsvath and ThurstonLOT18, Section 2]). Consider $\mathbf {x}\in \widehat {\mathit {BSA}}(\mathcal {H})$ and $\mathbf {x}'\in \widehat {\mathit {BSA}}(\mathcal {H}')$ so that

$$\begin{align*}m_{i+1}(\mathbf{x},a_1,\cdots,a_i)=0\quad\quad\text{and}\quad\quad m_{i+1}(\mathbf{x}',a_1,\cdots,a_i)=0\end{align*}$$

for all $i\ge 0$ and $a_i\in \mathcal {A}(\mathcal Z)$ . We say $\mathbf {x}$ and $\mathbf {x}'$ are equivalent if there exists a homotopy equivalence f from $\widehat {\mathit {BSA}}(\mathcal {H})$ to $\widehat {\mathit {BSA}}(\mathcal {H}')$ so that $f_1(\mathbf {x})=\mathbf {x}'$ and $f_{i+1}(\mathbf {x},a_1,\dots ,a_i)=0$ for all $i>0$ and $a_i\in \mathcal {A}(\mathcal Z)$ . Similarly, there is a type D homotopy equivalence from $\widehat {\mathit {BSD}}(\mathcal {H})$ to $\widehat {\mathit {BSD}}(\mathcal {H}')$ . If $\mathbf {x}\in \widehat {\mathit {BSD}}(\mathcal {H})$ and $\mathbf {x}'\in \widehat {\mathit {BSD}}(\mathcal {H}')$ are both closed elements (meaning that $\delta ^1(\mathbf {x})=\delta ^1(\mathbf {x}')=0$ ), we say $\mathbf {x}$ and $\mathbf {x}'$ are equivalent if there is a homotopy equivalence from $\widehat {\mathit {BSD}}(\mathcal {H})$ to $\widehat {\mathit {BSD}}(\mathcal {H}')$ that maps $\mathbf {x}$ to $\mathbf {x}'$ .

Given two bordered sutured manifolds $(M_1, \mathsf {\Gamma }_1, \mathcal Z)$ and $(M_2, \mathsf {\Gamma }_2, -\mathcal Z)$ , we may glue along $\mathcal Z$ to obtain a sutured manifold $(M, \Gamma )$ . Given $\beta $ -bordered sutured Heegaard diagrams $\mathcal {H}_1 = (\Sigma _1, {\boldsymbol {\alpha }}_1, {\boldsymbol {\beta }}_1, \mathcal Z)$ for $(M_1, \mathsf {\Gamma }_1, \mathcal Z)$ and $\mathcal {H}_2 = (\Sigma _2, {\boldsymbol {\alpha }}_2, {\boldsymbol {\beta }}_2, -\mathcal Z)$ for $(M_1, \mathsf {\Gamma }_1, -\mathcal Z)$ , there is a glued Heegaard diagram $\mathcal {H} = (\Sigma , {\boldsymbol {\alpha }}, {\boldsymbol {\beta }})$ , where $\Sigma = \Sigma _1 \bigcup _{Z} \Sigma _2$ ; the set ${\boldsymbol {\alpha }}$ is the union of ${\boldsymbol {\alpha }}_1$ and ${\boldsymbol {\alpha }}_2$ and the set ${\boldsymbol {\beta }}$ is the union of ${\boldsymbol {\beta }}_1$ and ${\boldsymbol {\beta }}_2$ , together with the circles formed by gluing the arcs in ${\boldsymbol {\beta }}_1^a$ and ${\boldsymbol {\beta }}_2^a$ along their endpoints on Z.

When at least one of the two diagrams is admissible, there is a gluing map

$$\begin{align*}\widehat{\mathit{BSA}}(\mathcal{H}_1) \boxtimes_{\mathcal{A}(\mathcal Z)} \widehat{\mathit{BSD}}(\mathcal{H}_2) \rightarrow \mathit{SFC}(\mathcal{H}) \end{align*}$$

which induces a chain homotopy equivalence of vector spaces. Chain homotopy equivalences of type A and type D modules induce chain homotopy equivalences of the box tensor product, so there is a well-defined equivalence class of $\mathbb F_2$ vector spaces

and an equivalence of chain homotopy equivalence classes of vector spaces

2.3.1 Supported contact structures

The construction of a compact manifold is natural from the data of a foliated open book: Pairs of successive pages define cornered cobordisms. As described in more detail below, we concatenate these to form a manifold with boundary and collapse the resulting components of $B\times I$ to circles and intervals again labeled B. The final page glues to the initial page via h. The resulting manifold M associated to the foliated open book $(\{S_i\}_{i=0}^{2k}, h)$ retains partial information about the sequence of cuts and adds in the abstract data. This information is encoded in the form of boundary decorations on $\partial M$ , and we introduce the kind of foliation we will consider before explaining how it arises.

See Figure 2.2 for local models of the integral curves.

Figure 2.2 Left: a hyperbolic singularity. Center: a positive elliptic singularity. Right: a regular foliation on an open set.

As is standard, leaves that enter (or exit) the hyperbolic singularities are called stable (or unstable) separatrices.

Each elliptic point e induces a cyclic order on the subset of hyperbolic points with separatrices terminating at e. If e is positive (respectively, negative), the order increases as the separatrices are encountered in a counterclockwise (clockwise) path around e.

Beginning with an abstract foliated open book, we now build a manifold whose boundary is naturally equipped with an ordered signed singular foliation. Each pair of successive pages defines an elementary cobordism $M_i$ from $S_{i-1}$ to $S_{i}$ with vertical boundary $(B\times I)\cup V_i$ , where $V_i$ is the union of a single saddle and of the collection of products $A_{i-1}\times I$ for any components $A_{i-1}$ that are left unchanged. More specifically, the saddle connects either the components of $A_{i-1}$ containing the endpoints of $\gamma _{i}$ or the component(s) containing $p_{i-1}$ and $q_{i-1}$ with the obtained component(s) of $A_i$ that have the same endpoints. See Figure 2.3.

Figure 2.3 The elementary cobordism associated to cutting $S_{i-1}$ along $\gamma _i$ , shown before collapsing $B\times I$ .

Concatenating these $M_i$ and gluing the final $S_{2k}$ to the initial $S_0$ via h yields a manifold with boundary $(B\times S^1) \cup V$ , where V is the (circular) union of the $V_i$ . The singular foliation on V has $2k$ singular points associated to the transitions between topological types of the pages, while the regular leaves may be identified with curves of the form $A_i\times \{t\}$ . If $S_{i-1}\xrightarrow {{\textbf {add}}} S_{i}$ , then the corresponding hyperbolic point $h_i$ is negative; otherwise, it is positive. We denote the set of hyperbolic points with respect to this partition $H=H_+\cup H_-$ . The signs match the partition of $[2k]$ introduced after Definition 2.4. Collapsing the components $B\times S^1$ to a single copy of B yields a manifold with foliated boundary $(M, \partial M, \mathcal {F})$ . (See Definition 2.9, below, for a precise definition of this term.) Each endpoint of B labeled $e_+$ becomes a positive elliptic point of the foliation, which we again denote by $e_+$ ; likewise, an endpoint $e_-$ becomes a negative elliptic point $e_-$ .

Example 2.8. We return to the foliated open book introduced in the first example. The associated smooth manifold is the solid torus shown in Figure 2.4. The images of $A_1$ and $A_2$ are shown in green and blue, respectively, on both the manifold on the left and in the associated boundary foliation on the right.

Figure 2.4 The smooth manifold (a solid torus) associated to the foliated open book from Figure 2.1 together with the foliation $\mathcal {F}$ on $\partial M$ .

In order to define the compatibility between contact structures and foliated open books, we examine the boundary foliation $\mathcal {F}$ more closely. To any smooth manifold constructed as above, we may associate an $S^1$ -valued Morse function $\pi \colon M \setminus B \rightarrow S^1$ whose level sets are the pages of the open book and which restricts to the boundary as an $S^1$ -valued Morse function ${\widetilde {\pi }}:=\pi |_{\partial M}\colon \partial M\setminus E\to S^1$ with the same critical points as $\pi $ . The function $\pi $ has only index 1 and 2 critical points, and all of these are index 1 critical points for ${\widetilde {\pi }}$ . The index with respect to $\pi $ is visible in the sign of these hyperbolic points: index 2 critical points of $\pi $ give positive hyperbolic points of the foliation, while index 1 critical points of $\pi $ give negative hyperbolic points of the foliation. Heuristically, this means that the interesting features of $\pi $ may all be seen from the foliation on the boundary. One may build a signed singular foliation simply by patching together the local models of Figure 2.2, but in the context of this paper we will only encounter ordered signed singular foliations; equivalently, these are signed singular foliations that can be induced by an $S^1$ -valued Morse function. We therefore consider the map ${\widetilde {\pi }}$ to be an essential part of the data and we write $\mathcal {F}=({\widetilde {\pi }}, H=H_-\cup H_+, E=E_-\cup E_+)$ . We may also assume that $0$ is a regular value of ${\widetilde {\pi }}$ , and we further require that ${\widetilde {\pi }}=\pm \phi $ on an $(r, \phi )$ -disk neighborhood of each elliptic point $e\in E$ , where the sign depends on the sign of the elliptic point. Here, $(r,\phi )$ is the standard polar coordinate system on $D^2$ . By construction, the foliation on a manifold constructed from an abstract foliated open book has only elliptic and four-pronged hyperbolic singularities, with signs inherited from the partitions of E and H.

We recall that these data include the $S^1$ -valued Morse function ${\widetilde {\pi }}$ , and we require that a diffeomorphism $f:(M_1, \partial M_1, \mathcal {F}_1) \rightarrow (M_2, \partial M_2, \mathcal {F}_2)$ between manifolds with foliated boundary must satisfy ${\widetilde {\pi }}_2\circ f= {\widetilde {\pi }}_1$ .

We have seen that an abstract foliated open book gives rise to a diffeomorphism class of manifolds with foliated boundary; we say a manifold with foliated boundary is compatible with $(\{S_i\}, h)$ if it is diffeomorphic to an output of this construction. A fixed manifold with foliated boundary will admit many different functions $\pi $ extending the Morse function ${\widetilde {\pi }}$ on the boundary, but compatibility identifies an equivalence class of such functions whose level sets are diffeomorphic to the pages $\{S_i\}$ .

Note that the serifed symbol $\Gamma $ used here is distinct from the sans-serif symbol $\mathsf {\Gamma }$ introduced in Section 2.2.2 as part of the defining data of a bordered sutured manifold. The above construction of $\Gamma (\mathcal {F})$ follows the construction of Giroux [Reference GirouxGir00] for dividing curves of characteristic foliations of convex surfaces and in our case too $\Gamma (\mathcal {F})$ indeed divides $\mathcal {F}$ : $\Gamma (\mathcal {F})$ is positively transverse to the leaves of $\mathcal {F}$ and it separates $\partial M\setminus \Gamma (\mathcal {F})$ into two parts, one of which contains all the positive singular points, while the other part contains all the negative singular points.Footnote ⁴ These properties in fact specify $\Gamma (\mathcal {F})$ up to isotopy transverse to $\mathcal {F}$ , and we will often work with its equivalence class, only specifying the representative when needed. The choice of the representative then changes $R_+(\mathcal {F})$ and $R_-(\mathcal {F})$ accordingly, while still preserving the condition that $\Gamma (\mathcal {F})=\partial R_+(\mathcal {F})=-\partial R_-(\mathcal {F})$ . Note that distinct foliations on $\partial M$ may induce isotopic dividing sets; see Figure 2.5.

Figure 2.5 Left: A foliation on $S^2$ with four elliptic points and two hyperbolic points. Right: A foliation on $S^2$ with two elliptic points. The dividing set for each foliation is a circle.

The term ‘topological’ refers to the fact that the above isotopy may not be smooth. In fact, in the next definition, the isotopy cannot be chosen to be smooth.

Distinct foliations may induce isotopic dividing sets, as illustrated in the following example.

2.3.2 Stabilization

Although each foliated open book supports a unique isotopy class of contact structures, there are many foliated open books which support the same class. Next, we define an operation which changes a foliated open book while preserving the contactomorphism class of the associated contact manifold.

In order to define stabilization along some arc in a page besides $S_0$ , we first describe an operation on foliated open books which preserves the contactomorphism type of the associated manifold.

Definition 2.18. [Reference Licata and VértesiLV20, Definition 3.16] The $1$ -shift of an abstract foliated open book $(\{S_i\}_{i=0}^{2k},h)$ is the foliated open book

$$\begin{align*}(\{S_i[1]\}_{i=0}^{2k},h[1])=(\{S_1,S_2,\dots,S_{2k},S_1'\},h'),\end{align*}$$

where

• - if $S_{0}\xrightarrow [\gamma _{1}]{{\textbf {cut}}} S_1$ , then $S_1'$ is defined by the relation $S_{2k}\xrightarrow [{h^{-1}(\gamma _{1})}]{{\textbf {cut}}} S_1'$ and $h'$ is the restriction of h to $S_1'$ ;

• - if $S_{0}\xrightarrow [{p_0, q_0}]{{\textbf {add}}} S_1$ , then $S_1'$ is defined by the relation $S_{2k}\xrightarrow [{ h^{-1}(p_0), h^{-1}(q_0) }]{{\textbf {add}}} S_1'$ and $h'$ is h extended by the identity on the added one-handle.

An r-fold iteration of the shift operation is called an r-shift and denoted by $(\{S_i[r]\}_{i=0}^{2k},h[r])$ . One can analogously define r-shifts for $r<0$ .

Shifts correspond to postcomposing $\pi $ with a diffeomorphism of $S^1$ .

It is easy to see that stabilizing does not change the underlying manifold with foliated boundary.

2.3.3 Sorted foliated open books

Next, we introduce some additional bookkeeping to record the cutting arcs and cocore arcs on all possible pages of the foliated open book. This additional information corresponds to a choice of a gradient-like vector field for $\pi $ on the associated smooth manifold, as we will describe in Section 2.3.4.

If $S_{{i-1}}\xrightarrow {{\textbf {add}}} S_{i}$ , then for all $j\geq {i}$ , decorate $S_{j}$ with a cocore of the added handle and label this cocore $\gamma _i^-$ . If $S_{{i-1}}\xrightarrow {{\textbf {cut}}} S_{i}$ along $\gamma \subset S_{{i-1}}$ , then for all $j\leq {i-1}$ , decorate $S_j$ with the cutting curve and label it $\gamma _i^+$ .

We call the $\gamma _i^\pm $ sorting arcs. Note that by an abuse of notation we use a single label $\gamma _i^\pm $ to refer to distinct copies of the sorting arcs on multiple $S_j$ . In general, sorting arcs may intersect (and consequently some sorting arcs may appear as a union of disconnected intervals on some pages). We will shortly restrict to the subclass of sorted foliated open books [Reference Licata and VértesiLV20, Section 5.3], which are characterized by the requirement that the sorting arcs are disjoint on all pages, together with a prescribed ordering of the intersections between sorting arcs and regular leaves. In the sorted case, we enhance the notation for describing a foliated open book to reflect these additional choices.

See Figure 2.7 in Example 2.24 for an illustration of the ordering conventions given above. Sorted foliated open books are useful because of their close relation to partial open books, which will be described in Section 2.3.5. Below, we show that a foliated open book need not be sorted.

Figure 2.6 The first three pages of the foliated open book from Example 2.5.

Figure 2.7 The stabilization arc $\gamma \subset S_0$ is chosen to have one endpoint on each interval of the binding B so that it crosses each of of the intersecting sorting arcs in Figure 2.6 exactly once. There is a right-handed Dehn twist identifying the two copies of the $S_2'$ page, while the monodromy $h' \colon \thinspace S_4'\to S_0'$ is translation in the plane of the page. The sorting arcs $\gamma ^+$ are shown in light blue, and the sorting arcs $\gamma ^-$ are shown in dark blue. Subintervals $I_\pm \subset A_0$ are highlighted in green and orange on $S_0$ and may be chosen analogously on the other pages.

We may always stabilize a foliated open book to obtain a sorted one.

2.3.4 Gradient flows on foliated open books

As in Section 2.3.1, we consider the smooth manifold with foliated boundary $(M, \partial M, \mathcal {F})$ constructed from a foliated open book. The initial abstract data determines an equivalence class of circle-valued Morse functions $\pi \colon \thinspace M\setminus B\rightarrow S^1$ [Reference Licata and VértesiLV20, Section 5.2]. The level sets of $\pi $ are the pages, and the restriction ${\widetilde {\pi }} \colon \thinspace \partial M\setminus B \rightarrow S^1$ is again a circle-valued Morse function with the same set of critical points. In this setting, one may consider the critical submanifolds of a gradient-like vector field for $\pi $ parallel to $\partial M$ , which thus restricts to a gradient-like vector field for ${\widetilde {\pi }}$ . The intersection of the critical submanifolds with the regular level sets of $\pi $ determines sorting arcs on the pages of the corresponding abstract foliated open book [Reference Licata and VértesiLV20, Section 5.3]. Thus, a foliated open book is sorted when a gradient-like vector field on $\partial M$ that satisfies the ordering conditions for sorting arcs extends to a gradient-like vector field on M whose critical submanifolds are disjoint.

Any surface with an open book foliation decomposes along regular leaves into square tiles that each contain precisely one hyperbolic singularity. Figure 2.8 shows the flowlines of a preferred $\nabla {\widetilde {\pi }}$ on such a tile; these flowlines satisfy the ordering conditions in Definition 2.21, although ensuring their extension to M yields a sorted foliated open book may require stabilization.

Figure 2.8 The tile of $\mathcal {F}$ shows purple flowlines of $\nabla {\widetilde {\pi }}$ which satisfy the sorted ordering conditions.

The Morse perspective is useful not only for building geometric intuition but also for relating the boundary decorations to those in the interior of the manifold. For example, a sorting arc $\gamma ^{+}_i$ is isotopic through the associated stable critical submanifold of $\nabla \pi $ to the stable submanifold of $\nabla {\widetilde {\pi }}$ on $\partial M$ which passes through the corresponding positive hyperbolic point $h_i$ . Likewise, a sorting arc $\gamma ^{-}_i$ is isotopic through the associated unstable critical submanifold of $\nabla \pi $ to the unstable submanifold of $\nabla {\widetilde {\pi }}$ on $\partial M$ which passes through the corresponding negative hyperbolic point $h_i$ . Conditions 1, 2 and 3 in Definition 2.21 are chosen so that the graphs formed by the (un)stable separatrices of positive (negative) hyperbolic points with respect to $\mathcal {F}$ and with respect to $\nabla {\widetilde {\pi }}$ are isotopic. See [Reference Licata and VértesiLV20, Lemma 8.8]. In the next section, this will allow us to define certain subsurfaces as neighborhoods of either graph.

2.3.5 Sufficiently stabilized foliated open books and the associated partial open book

Sorted foliated open books are closely related to partial open books, and [Reference Licata and VértesiLV20] introduced the technical designation of a ‘sufficiently stabilized’ foliated open book. In this section, we recall this definition and show that a sufficiently stabilized foliated open book naturally has a companion partial open book. Throughout this section, we will let $(\{S_i\}_{i=0}^{2k},h,\{\gamma _i^\pm \})$ be a sorted foliated open book and $(M, \xi , \mathcal {F})$ be the supported foliated contact three-manifold. We further assume that either $k>0$ or $|\partial M|>1$ ; the only foliated open books this excludes are those formed from an honest open book by removing a neighborhood of a single point on the binding.

Set

$$\begin{align*}R_i=N_{\lrcorner}\big(A_i \cup\big(\bigcup_{i < j}\gamma_j^+\big)\cup\big(\bigcup_{i\geq j}\gamma_j^-\big)\big) \subset S_i,\\[-17pt]\end{align*}$$

where $N_{\lrcorner }$ denotes a ‘cornered’ neighborhood of $A_i \cup (\bigcup _{i < j}\gamma _j^+)\cup (\bigcup _{i\geq j}\gamma _j^-)$ , with corners at E so that $R_i$ meets B only at E. Furthermore, define

$$\begin{align*}P_i=\overline{S_i\setminus R_i},\end{align*}$$

as shown on the left in Figure 2.9. Since only the $R_i$ change with i, we can identify all the $P_i$ with each other by the flow of the gradient-like vector field. We denote the composition of these identifications from $P_0\subset S_0$ onto $P_{2k}\subset S_{2k}$ by $\iota $ . Let $\widetilde {S}=S_{0}$ , $\widetilde {P}=P_{0}$ and $\tilde {h}=h\vert _{P_{2k}}\circ \iota $ .

Figure 2.9 The shaded regions show the cornered neighborhoods $R_i''$ in each page of the sufficiently stabilized open book. As in Example 2.24, successive pages are identified by translation except for the right-handed Dehn twist $\tau $ at $S_2''$ .

This condition ensures that $(\widetilde {S}, \widetilde {P}, \tilde {h})$ is a partial open book whose pages embed into the pages of $(\{S_i\}_{i=0}^{2k}, h, \{\gamma _i\})$ . As the term suggests, this condition may always be achieved by a sequence of positive stabilizations, and in fact, these may be chosen to occur on the $S_0$ page.

As shown in [Reference Licata and VértesiLV20, Lemma 8.12], the cornered diffeomorphism type of the subsurfaces $R_i\subset S_i$ depends only on the foliation $\mathcal {F}$ , rather than on the pages $S_i$ . Construct the corresponding contact three-manifold $(M,\xi ,\mathcal {F})$ for a sorted abstract foliated open book $(\{S_i\},h,\{\gamma _i^\pm \})$ . Recall from Definition 2.10 that the subsurface $R_+(\mathcal {F})$ (respectively $R_-(\mathcal {F})$ ) of $\partial M$ is a closed neighborhood of the (un)stable separatrices corresponding to positive (negative) hyperbolic points of $\mathcal {F}$ .

As noted in Section 2.3.4, the graph of positive separatrices of $\nabla {\widetilde {\pi }}$ from positive hyperbolic points on $\partial M$ is (nonsmoothly) isotopic to the graph of positive separatrices of $\mathcal {F}$ from positive hyperbolic points on $\partial M$ . On the other hand, the former is isotopic through the stable critical submanifolds to a neighborhood of the sorting arcs $\cup _{H_+}\gamma ^+_i$ on $S_0$ . This yields an identification between $R_+(\mathcal {F})\subset \partial M$ and $R_0\subset S_0$ . Similarly, studying negative separatrices of negative hyperbolic points and unstable critical submanifolds yields an identification between $R_-(\mathcal {F})\subset \partial M$ and $R_{2k}\subset S_{2k}$ .

Similarly, all the $R_i$ can be identified with surfaces described only using the foliation $\mathcal {F}$ , independently of the foliated open book, as follows. Recall that $R_i$ is the (cornered) neighborhood of the union of $A_i$ and the intersection of all stable and unstable submanifolds of $\nabla \pi $ with $S_i$ . Each of these intersections can be pushed up (or down) onto $\partial M$ along the corresponding stable (or unstable) submanifolds. While performing these isotopies we push the half-neighborhood of $A_i$ up along $I_+$ and down along $I_-$ ; the surface obtained after the isotopy is not a submanifold of $\partial M$ , as it has a half twist in the middle of each component I of $A_i$ which extends into the interior of M. See Figure 2.10.

Figure 2.10 The blue region on the left-hand picture is the cornered neighborhood $R_i\subset S_i$ . On the right-hand figure, $R_i$ has been isotoped through M in the direction of the indicated arrows to lie mostly on $\partial M$ . The small arrow indicates the half twist which extends into the interior of M. Note that B has been blown up to $B\times I$ for clarity.

2.3.6 Gluing foliated open books

In order to glue foliated open books, we describe the matching conditions imposed upon their boundary foliations. Recall that a foliation $\mathcal {F}$ on a surface F always refers to the data of a signed foliation together with the function ${\widetilde {\pi }}\colon F\setminus E\to S^1$ whose level sets are the leaves of $\mathcal {F}$ .

The main advantage of foliated open books is that they are well suited for gluing. Suppose that $(M^L,\xi ^L,\mathcal {F}^L)$ and $(M^R,\xi ^R,\mathcal {F}^R)$ are foliated contact three-manifolds such that there exists an orientation-reversing diffeomorphism $\psi \colon \partial M^L \to \partial M^R$ that maps the reverse of the foliation $\mathcal {F}^L$ onto $\mathcal {F}^R$ .

Recall that by hypothesis, the boundary foliations are divided, so the boundaries $\partial M^{\bullet }$ are convex with respect to $\xi ^{\bullet }$ . (Here, and throughout this section, $\bullet $ will be an element of the set $\{R, L\}$ .) Since the contact structures are I-invariant near the boundaries, $\psi $ determines a closed contact three-manifold

$$\begin{align*}(M,\xi)=(M^L\cup_\psi M^R,\xi^L\cup_\psi\xi^R).\end{align*}$$

If the initial contact manifolds were supported by foliated open books $(\{S^L_i\},h^L)$ and $(\{S^R_i\},h^R)$ , respectively, then $(M,\xi )$ naturally inherits a supporting open book whose pages and binding are built by gluing the pages and bindings of the constituent pieces. More precisely, recall that $\partial S^{\bullet }_i=A_i^{\bullet }\cup B^{\bullet }$ , where all the $B^{\bullet }$ are identified when forming the manifolds $M^{\bullet }$ . The map $\psi $ identifies the intervals $A_i^L$ and $-A_i^R$ for all i, forming the surfaces

$$\begin{align*}S_i=S_i^L\bigcup_{A_i^L\xrightarrow{\psi}-A_i^R} S_i^R\end{align*}$$

with boundary $B=B^L\cup B^R$ . Observe that a cut to $S_i^R$ pairs with a handle addition to $S_i^L$ , and vice versa, so the surfaces $S_i$ are diffeomorphic for all i. This allows us to identify $S_0\cong S_1 \cong \cdots \cong S_{2k}$ , and we denote the composition of these identifications by $\iota \colon S_0\to S_{2k}$ . Note that $\iota $ fixes B, and it restricts to $P_{0}^\bullet $ as the identification $\iota ^\bullet :P_0^\bullet \rightarrow P_{2k}^\bullet $ defined in Section 2.3.5. Letting $S:=S_0$ , the monodromy $h\colon S\to S$ for the glued-up open book can be obtained as the composition of $\iota $ with $h^L\cup h^R\colon S_{2k}=S_{2k}^L\cup S_{2k}^R\to S_{0}^L\cup S_{0}^R=S_0$ . By construction, h fixes B, as required.

Since $\iota \colon S_0\to S_{2k}$ is given as a sequence of identifications, the monodromy h may be difficult to reconstruct in a complicated case, and the above construction does not automatically give a factorization in terms of Dehn twists. For sorted open books, however, there is a straightforward way to describe $\iota $ (and thus h) as follows. Suppose now, that $(\{S^L_i\},h^L, \{\gamma _i^{\pm ,L}\})$ and $(\{S^R_i\},h^R,\{\gamma _i^{\mp ,R}\})$ are sorted foliated open books compatible with $(M^L,\xi ^L,\mathcal {F}^L)$ and $(M^R,\xi ^R,\mathcal {F}^R)$ , respectively. Recall that each page $S_i^{\bullet }$ decomposes as the union of a ‘constant’ part $P_i^{\bullet }$ and an i-dependent part $R_i^{\bullet }$ which is determined by the foliation $\mathcal {F}^{\bullet }$ . By construction, $P_0^{\bullet }\cong P_1^{\bullet }\cong \cdots \cong P_{2k}^{\bullet }$ . These diffeomorphisms are explicit in the construction of the pages $S_i^{\bullet }$ , and together they determine $\iota \vert _{P_0^L\cup P_0^R}=\iota ^L\cup \iota ^R\colon P_0^L\cup P_0^R\to P_{2k}^L\cup P_{2k}^R$ .

As described earlier, sorted foliated open books have the property that $R_0^L$ is isotopic in $M^L$ to $R_+(\mathcal {F}^L)$ . This surface is mapped by $\psi $ onto $R_-(\mathcal {F}^R)$ , which in turn is isotopic in $M^R$ to $R_{2k}^R$ . The composition of these three maps gives $\iota \vert _{R_0^L}\colon R_0^L\to R_{2k}^R$ , and we can similarly obtain $\iota \vert _{R_0^R}\colon R_0^R\to R_{2k}^L$ . This yields a concrete description of the entire map $\iota $ on the remaining parts $R_{2k}^L\cup R_{2k}^L$ . Together, the above maps determine h everywhere.