Partial Open Book in Contact Topology

• Partial open book is a structure defined by the triple (S, P, h) that decomposes a contact 3-manifold with convex boundary using handle attachments and prescribed boundary identifications.
• It employs Morse-theoretic techniques and balanced sutured manifold constructions to establish a relative Giroux correspondence in contact geometry.
• The framework supports extendable monodromy and computational adaptations, linking classical open book theory with novel applications including restricted open-book question answering.

In contact topology, a partial open book is a relative analogue of an ordinary open book for a contact $3$-manifold with convex boundary. In the formulation used by Etgü–Ozbağcı and developed further for Morse-theoretic and foliation-theoretic purposes, it is encoded by a triple $(S,P,h)$ in which $S$ is a compact connected oriented surface with nonempty boundary, $P\subseteq S$ is a subsurface built by $1$-handles, and $h:P\to S$ is a homeomorphism onto its image that is the identity on $A=\partial P\cap\partial S$ (Licata et al., 2017). In a distinct research usage, the phrase also denotes a restricted open-book question-answering regime in which only a narrow slice of source text, typically one retrieved paragraph, is available at test time (Ciosici et al., 2021).

1. Abstract data and the associated sutured manifold

A partial open book begins with a surface pair $(S,P)$. The handle-theoretic condition used in the Morse-structure literature is that $(S,P)$ admit a compatible handle structure: there are $1$-handles $(S,P,h)$0 in $(S,P,h)$1 such that

$(S,P,h)$2

and $(S,P,h)$3 is obtained from

$(S,P,h)$4

by successively attaching those $(S,P,h)$5-handles (Licata et al., 2017). The resulting abstract partial open book is the triple

$(S,P,h)$6

where

$(S,P,h)$7

is a homeomorphism onto its image and $(S,P,h)$8 is the identity on $(S,P,h)$9. The extreme cases $S$0 and $S$1 are allowed; when $S$2, one recovers an ordinary abstract open book (Licata et al., 2017).

The associated $S$3-manifold is built from two pieces,

$S$4

and

$S$5

glued by identifying $S$6 and imposing

$S$7

The result is denoted

$S$8

Its boundary is

$S$9

and the sutures are

$P\subseteq S$0

The positive and negative regions are

$P\subseteq S$1

Since $P\subseteq S$2 is a homeomorphism,

$P\subseteq S$3

so $P\subseteq S$4 is a balanced sutured manifold (Licata et al., 2017).

2. Contact-geometric role and relative Giroux correspondence

The contact interpretation is relative rather than closed. The pieces $P\subseteq S$5 and $P\subseteq S$6 carry unique tight contact structures with dividing sets $P\subseteq S$7 on $P\subseteq S$8 and $P\subseteq S$9 on $1$0, and these glue to a contact structure on $1$1. A contact manifold with convex boundary $1$2 is said to be presented by $1$3 when it is contactomorphic to $1$4 with this glued contact structure (Licata et al., 2017).

Isomorphism of partial open books is defined by a diffeomorphism

$1$5

such that $1$6 and

$1$7

With this notion, the standard relative Giroux correspondence identifies isomorphism classes of partial open books, up to positive stabilization, with compact contact $1$8-manifolds with convex boundary, up to contactomorphism (Licata et al., 2017).

This places partial open books between two familiar theories. They are a relative analogue of ordinary open books for closed contact manifolds, but they are adapted to convex boundary and sutures rather than to a closed binding. A plausible implication is that their natural invariants are boundary-sensitive from the outset, whereas ordinary open books recover boundary data only after cutting.

3. Extendable monodromy, Morse structures, and Legendrian front theory

A central refinement is the notion of extendable monodromy. If $1$9 and $h:P\to S$0 restricts to $h:P\to S$1, then $h:P\to S$2 extends $h:P\to S$3. The monodromy $h:P\to S$4 is called extendable if it extends to a homeomorphism

$h:P\to S$5

with

$h:P\to S$6

In that case there is a contact embedding

$h:P\to S$7

so the partial-open-book manifold sits as a contact submanifold of a closed open-book manifold (Licata et al., 2017). Extendability is nontrivial: a necessary condition is

$h:P\to S$8

and a sufficient condition stated in the paper is that both $h:P\to S$9 and $A=\partial P\cap\partial S$0 are connected (Licata et al., 2017).

Under this hypothesis one can import Gay–Licata’s Morse-structure technology. A Morse structure function for $A=\partial P\cap\partial S$1 is a smooth map

$A=\partial P\cap\partial S$2

such that $A=\partial P\cap\partial S$3, each page carries a Morse function with finitely many index-$A=\partial P\cap\partial S$4 critical points and no index-$A=\partial P\cap\partial S$5 critical points, the family is Morse–Smale away from isolated handleslide values, and

$A=\partial P\cap\partial S$6

after identifying the ends (Licata et al., 2017). A Morse structure on $A=\partial P\cap\partial S$7 is such an $A=\partial P\cap\partial S$8 together with a vector field $A=\partial P\cap\partial S$9 tangent to pages, gradient-like on each page, and standard near the binding. Every partial open book with extendable monodromy admits a Morse structure (Licata et al., 2017).

The associated Morse diagram is built from the family of page slices and records the evolution of co-cores, primary pairs, antecedent pairs, and markers. The resulting contact-geometric normal form is explicit: after isotoping the supported contact structure through contact structures presented by $(S,P)$0, the interior of each component of

$(S,P)$1

is contactomorphic to a contact submanifold of

$(S,P)$2

This model supports a front-projection theory for properly embedded Legendrian tangles. If a Legendrian tangle is disjoint from the binding and transverse to the skeleton, its front on the Morse diagram determines it completely, and Legendrian isotopy is generated by the Gay–Licata moves together with the boundary moves $(S,P)$3, $(S,P)$4, and $(S,P)$5 (Licata et al., 2017).

4. Foliated open books as a finer boundary-sensitive refinement

Foliated open books were introduced as a new type of open book decomposition for a contact $(S,P)$6-manifold with a specified characteristic foliation $(S,P)$7 on its boundary. The paper defines three versions—embedded, Morse, and abstract—and proves their equivalence, together with existence, uniqueness, and a Giroux correspondence for the triple $(S,P)$8 (Licata et al., 2020).

Their relation to partial open books is explicit and asymmetric. A foliated open book records the actual boundary characteristic foliation, whereas a partial open book is organized around convex boundary and dividing-set data. The paper therefore states that foliated open books are a finer tool than partial open books because the boundary foliation carries more data than the dividing set (Licata et al., 2020).

The comparison theorems are stabilization-dependent. Any sufficiently positively stabilized foliated open book for $(S,P)$9 contains a partial open book for $(S,P)$0 as a submanifold with page-wise inclusions, and any sufficiently positively stabilized partial open book may be obtained this way (Licata et al., 2020). At the same time, the passage from partial to foliated is non-unique: a given partial open book may give rise to several non-equivalent foliated open books (Licata et al., 2020).

This refinement has practical consequences. Foliated open books have user-friendly cutting and gluing properties and arise naturally as submanifolds of classical open books for closed $(S,P)$1-manifolds. The paper emphasizes that gluing is possible with partial open books but is less straightforward, precisely because dividing-set data are coarser than boundary-foliation data (Licata et al., 2020).

5. Adjacent frameworks, computational tools, and higher-dimensional limits

Several neighboring frameworks illuminate what partial open books do and do not capture. In the computational direction, recent Heegaard Floer software for closed contact $(S,P)$2-manifolds starts from an abstract open book $(S,P)$3 together with a collection

$(S,P)$4

of pairwise disjoint properly embedded arcs on $(S,P)$5 cutting it into disks, and then forms the multipointed Heegaard diagram

$(S,P)$6

That work does not develop partial open books explicitly, but it identifies arc systems, region combinatorics, distinguished contact generators, and nice-diagram reductions as the most transferable ingredients toward a sutured or partial-open-book adaptation (Kutluhan et al., 4 Feb 2025).

In another adjacent direction, spinal open books are introduced as a generalization of ordinary open books on closed $(S,P)$7-manifolds, with

$(S,P)$8

where the binding is replaced by a spine that is an $(S,P)$9-bundle over compact surfaces with boundary. The relevant paper explicitly presents spinal open books as a generalization of ordinary open books, not as a reformulation of partial open books, though it uses relative open books and manifolds with boundary in intermediate constructions (Min et al., 2024).

Higher-dimensional work is mostly negative from the standpoint of universality. Recent $1$0-manifold results prove that there exist infinitely many parallelizable closed oriented $1$1-manifolds that do not admit open book decompositions, and derive that there are infinitely many Engel manifolds admitting no supporting open book (Lawande et al., 17 Sep 2025). Those papers are explicit that their direct theorems concern only closed $1$2-manifolds and do not develop a theory of partial open books in the sutured/contact-boundary sense (Lawande et al., 17 Sep 2025, Kastenholz, 15 Apr 2025). This suggests that any genuinely $1$3-dimensional partial-open-book theory would have to be broader than the classical closed open-book paradigm.

6. Alternate usage in restricted open-book question answering

Outside topology, “partial open book” is used informally for a restricted retrieval regime in question answering over long instructional documents. The LEFT benchmark introduces three conditions—prior knowledge only, closed-book after reading, and open-book—and the open-book condition gives the model access not to the full textbook but to one retrieved natural paragraph concatenated with the statement to be judged true or false (Ciosici et al., 2021).

In that setting, the accessible evidence is intentionally narrow. The statement-plus-paragraph input is truncated to $1$4 word pieces, retrieval is performed either automatically using sBERT or by supplying a manually identified relevant paragraph (“goldIR”), and the open-book setting is therefore naturally interpretable as a restricted open-book or partial open-book formulation (Ciosici et al., 2021). The benchmark’s results show the distinction sharply: parametric closed-book performance is essentially random on the balanced task, one automatically retrieved paragraph improves accuracy to roughly $1$5–$1$6, and even the gold paragraph yields only about $1$7–$1$8 (Ciosici et al., 2021).

This usage is terminologically separate from the contact-topological notion. The shared idea is limited access to a larger book-like structure, but the mathematical partial open book is a decomposition of a contact $1$9-manifold with convex boundary, whereas the QA usage denotes restricted test-time access to external evidence.