Spinal Open Book Decompositions

• Spinal open book decompositions are a framework that splits a 3-manifold into spine and paper regions, encoding contact structures and their symplectic/Stein fillings.
• They utilize Lefschetz fibrations and mapping class group techniques to classify fillings through monodromy factorizations and holomorphic curve foliations.
• Spine removal surgery and planar torsion obstructions provide universal topological bounds and fillability criteria, extending classical open book methods.

A spinal open book decomposition is a generalization of the classical open book decomposition for 3-manifolds, introduced to provide a flexible framework for encoding contact structures compatible with symplectic and Stein fillings, particularly in the context of Lefschetz fibrations over surfaces with boundary. The notion of spinal open books enables a precise description of the boundary behavior of such fillings and provides new methods for constructing and obstructing symplectic fillings by leveraging holomorphic curve theory and mapping class group techniques. This structure is central to the study of planar contact manifolds and the geography of their symplectic fillings, as it naturally incorporates phenomena—such as positive multisections and exotic singularities—not seen in the classical setting.

1. Definition and Structure of Spinal Open Book Decompositions

A spinal open book decomposition of a compact oriented 3-manifold $M$ (possibly with boundary) is a decomposition

$M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$

where the interiors of $M_{\mathrm{spine}}$ and $M_{\mathrm{paper}}$ are disjoint. The structure is specified by:

• Spine ($M_{\mathrm{spine}}$): There is a compact, oriented surface $\Sigma$ (possibly disconnected, called the "vertebrae") with nonempty boundary, and a trivial $S^1$-bundle $$\pi_{\mathrm{spine}}\colon M_{\mathrm{spine}} \cong \Sigma \times S^1 \to \Sigma$$. The fibers are $S^1$.
• Paper ($M_{\mathrm{paper}}$): This admits a mapping torus description $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$0 where $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$1 is a compact oriented surface with nonempty boundary, and $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$2 is the monodromy. Each fiber of the bundle $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$3 (the "pages") is such a surface.
• Interface: Along each torus boundary component $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$4, the circle fibers of $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$5 coincide with boundary components of pages of $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$6. For each $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$7, there is a preferred "meridian" class $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$8 in $M = M_{\mathrm{spine}} \cup M_{\mathrm{paper}}$9; together with the class $M_{\mathrm{spine}}$0 of a page-boundary circle, the pair $M_{\mathrm{spine}}$1 gives a positively oriented basis.

A contact structure $M_{\mathrm{spine}}$2 is said to be supported by a spinal open book if:

• $M_{\mathrm{spine}}$3 is positive on each page,
• the Reeb vector field $M_{\mathrm{spine}}$4 is tangent to the $M_{\mathrm{spine}}$5-fibers of the spine,
• on the boundary, $M_{\mathrm{spine}}$6 is tangent to the page fibers, and the characteristic foliation on each interface torus consists of closed leaves in the meridian class $M_{\mathrm{spine}}$7 (Lisi et al., 2018, Lisi et al., 2020, Min et al., 2024).

2. Relationship to Lefschetz Fibrations and Fillings

A bordered Lefschetz fibration $M_{\mathrm{spine}}$8 (over a compact oriented surface $M_{\mathrm{spine}}$9 with boundary) induces a spinal open book on its boundary:

• The "spine" is given by the horizontal boundary $M_{\mathrm{paper}}$0 (an $M_{\mathrm{paper}}$1-bundle over $M_{\mathrm{paper}}$2).
• The "paper" is given by the vertical boundary $M_{\mathrm{paper}}$3 (fibered over $M_{\mathrm{paper}}$4).

Allowable Lefschetz fibrations (critical points with nonhomologically trivial vanishing cycles) naturally yield spinal open books, and the fillability properties, as well as the possible symplectic and Stein structures, are reflected in the geometry and monodromy data of these fibrations.

Explicitly, for planar spinal open books (those where at least one page is genus zero), there is a classification:

• Minimal strong/Stein/Weinstein fillings correspond, up to symplectic/Weinstein deformation, to allowable bordered Lefschetz fibrations with prescribed boundary spinal data. This establishes a bijection between the equivalence classes of fillings and that of Lefschetz fibrations intertwining the spinal open book structure (Lisi et al., 2020, Min et al., 2024).

3. Holomorphic Curve Foliations, Exotic Singularities, and Their Role

A central technique for the analysis of fillings is the use of holomorphic curve theory:

• For $M_{\mathrm{paper}}$5 admitting a uniform Lefschetz-amenable spinal open book with planar pages, any filling $M_{\mathrm{paper}}$6 can be "completed" with an almost complex structure $M_{\mathrm{paper}}$7 to admit a foliation by punctured $M_{\mathrm{paper}}$8-holomorphic curves.
• The leaves of this foliation are of three types: regular (homeomorphic to pages), ordinary singular (nodal unions), and exotic fibers associated to boundary-interchange (half-twist) singularities of the underlying fibration.

Local models for exotic singularities are given by projections such as $M_{\mathrm{paper}}$9, $M_{\mathrm{spine}}$0, where the regular fibers are pair-of-pants surfaces and the exotic fiber at $M_{\mathrm{spine}}$1 reflects the boundary-interchange monodromy (Min et al., 2024).

The compactness, intersection, and index calculations for these holomorphic curves, together with mapping class group techniques, lead to the uniqueness and classification statements for fillings, as well as the construction of symplectic cobordisms (see also "spine removal surgery" below) (Lisi et al., 2020).

4. Spine Removal Surgery and Universal Topological Bounds

Spine removal surgery is a symplectic cobordism technique crucial for both obstruction and classification results. Given a spinal open book domain $M_{\mathrm{spine}}$2 with spinal submanifold $M_{\mathrm{spine}}$3 (for a subsurface $M_{\mathrm{spine}}$4), the surgery consists in:

• Forming the cobordism $M_{\mathrm{spine}}$5, attaching symplectic handles along the spine.
• Capping off the page-boundaries adjacent to $M_{\mathrm{spine}}$6 with disks, resulting in a new 3-manifold $M_{\mathrm{spine}}$7 and spinal open book (Lisi et al., 2018, Lisi et al., 2019).

This operation transforms the analysis of fillings into a problem on a closed symplectic 4-manifold, allowing the use of Lefschetz fibration structures and holomorphic sphere foliations to obtain:

• Universal bounds on topological invariants: If $M_{\mathrm{spine}}$8 is supported by a symmetric planar spinal open book, then any minimal strong symplectic filling $M_{\mathrm{spine}}$9 satisfies $\Sigma$0, with $\Sigma$1 depending only on the spinal open book data.
• The argument leverages the construction $\Sigma$2, which is diffeomorphic to $\Sigma$3, with genus, Euler characteristic, and the number of exceptional spheres all bounded in terms of the spinal data (Lisi et al., 2019).

5. Monodromy, Mapping Class Groups, and Classification

Spinal open books introduce new monodromy phenomena compared to the classical case, captured by the spinal mapping class group $\Sigma$4, generated by:

• Interior diffeomorphisms of the page $\Sigma$5 (fixing the boundary),
• Boundary-interchange half-twists $\Sigma$6 along arcs connecting pairs of boundary components.

Classifying fillings boils down to understanding positive admissible factorizations in $\Sigma$7:

• Total monodromy is written as a product of boundary-interchange twists (corresponding to branch points or exotic fibers) and Dehn twists (Lefschetz singularities).
• Equivalence of fillings corresponds to Hurwitz-equivalence of such factorizations.
• In particular, there is a one-to-one correspondence between deformation classes of minimal strong fillings and equivalence classes of these monodromy factorizations (Min et al., 2024).

This framework allows classification of fillings in settings where the classical Giroux open book approach is insufficient, such as for torus bundles, certain non-orientable circle bundles, and contact structures with no Giroux torsion (Lisi et al., 2020, Min et al., 2024).

6. Obstructions and Invariants: Planar $\Sigma$8-torsion and ECH/SFT

Spinal open books also reveal new fillability obstructions and inform contact invariants:

• Planar $\Sigma$9-torsion: If a domain has a planar piece (page) with $S^1$0 boundary components, but the spinal open book is not symmetric, (i.e., different vertebrae are not hit equally), then strong symplectic fillability is obstructed. If additionally $S^1$1 (a closed 2-form) is exact on all affected spinal components, so-called $S^1$2-separating, even weak fillability can be ruled out. Examples include many torus bundles with annular pages (Lisi et al., 2018).
• ECH and SFT torsion: The presence of a planar $S^1$3-torsion domain annihilates the ECH contact class with certain twisted coefficients and forces algebraic $S^1$4-torsion in Symplectic Field Theory, providing new algebraic fillability obstructions (Lisi et al., 2020).

These obstruction phenomena often rely on spine removal, holomorphic curve degenerations, or explicit combinatorial arguments in the mapping class group.

7. Applications and Further Examples

Spinal open books subsume the classical planar open books as a special case (single spine component), but also capture more general situations:

• Circle bundles over surfaces, including non-orientable cases, admit spinal open books whose page and monodromy encode $S^1$5-invariant contact structures. The classification of Stein and strong fillings for these, as well as concrete fillability obstructions, follows from the analysis above (Lisi et al., 2020, Lisi et al., 2018).
• Torus bundles: Parabolic, elliptic, and non-orientable torus bundles have their strong and Stein fillings completely classified by monodromy factorizations in $S^1$6, with explicit existence and uniqueness results (Min et al., 2024).
• Lens spaces and high-genus examples: Spinal open books reflect the subtlety of Stein filling types and topological invariants, often expressing them in terms of geometric monodromy data and mapping class group relations such as lantern relations (Lisi et al., 2019).

A significant contribution of recent work is the demonstration that many contact 3-manifolds, even those admitting high-genus open book structures, can equivalently be described via a planar spinal open book, broadening the reach of these classification and obstruction theorems (Min et al., 2024).