Spinal Open Books: A Contact Framework

• Spinal open books are decompositions of 3-manifolds where the binding is replaced by S¹-bundles over surfaces, extending the classical open book model.
• They support unique contact structures via compatible Giroux forms and enable symplectic and Stein fillings through Lefschetz fibrations over non-disk bases.
• Planar spinal open books yield finiteness results for filling invariants, while higher-dimensional variants produce infinitely many Weinstein fillings using representation theory.

A spinal open book is a generalization of an ordinary open book decomposition in which the binding is replaced by a “spine” built from $S^1$-bundles over surfaces, while the complement remains fibred by pages. In dimension three, this replaces the usual binding $\coprod S^1\times D^2$ by pieces of the form $S^1\times \Sigma_i$, where each $\Sigma_i$ is a compact surface with boundary, and it provides the natural boundary structure for Lefschetz fibrations over compact oriented surfaces with boundary rather than only over the disk (Kaloti, 2013, Lisi et al., 2018). Subsequent work has developed spinal open books into a contact-topological framework with existence and uniqueness of compatible contact structures, filling theorems for planar cases, a monodromy theory involving a spinal mapping class group, and higher-dimensional constructions of manifolds with infinitely many Weinstein fillings (Lisi et al., 2018, Min et al., 2024, Zhou, 2023).

1. Definition and topological structure

In the form used in the filling literature, a spinal open book decomposes a closed oriented 3-manifold $M$ into a spine and a paper. One standard formulation writes

$M = M_\Sigma \cup M_P,$

where $M_\Sigma$ is fibred by circles over a compact oriented surface $\Sigma$ with nonempty boundary, and $M_P$ is a mapping torus over $S^1$ whose fibers are compact oriented surfaces with nonempty boundary; the page boundaries coincide with circle fibers of the spine (Lisi et al., 2018, Min et al., 2024). In dimension three, the spine bundle is trivial, so each connected spine component is diffeomorphic to $\coprod S^1\times D^2$0 (Lisi et al., 2018, Min et al., 2024).

A closely related abstract description specifies a 5-tuple

$\coprod S^1\times D^2$1

where $\coprod S^1\times D^2$2 is the union of pages, $\coprod S^1\times D^2$3 is an orientation-preserving diffeomorphism of $\coprod S^1\times D^2$4 fixing $\coprod S^1\times D^2$5 pointwise, $\coprod S^1\times D^2$6 is the spine, and

$\coprod S^1\times D^2$7

is a bijection describing how boundary components of pages are attached to boundary components of the spine (Kaloti, 2013). This formulation makes explicit that pages may be disconnected, different page components may be non-diffeomorphic, and distinct spine components may also have different topology (Kaloti, 2013).

Near interface tori, the decomposition carries local coordinates adapted simultaneously to the paper fibration and the spine fibration. One paper-side model uses coordinates $\coprod S^1\times D^2$8 with

$\coprod S^1\times D^2$9

where $S^1\times \Sigma_i$0 is the multiplicity along that boundary component; the corresponding spine-side model uses coordinates $S^1\times \Sigma_i$1 with

$S^1\times \Sigma_i$2

and matching torus coordinates on the overlap (Min et al., 2024). Earlier geometric treatments also encode multiplicity as the number of distinct boundary components of a page lying in a given boundary torus (Lisi et al., 2018).

Classical Giroux open books appear as the special case in which every vertebra is a disk and all multiplicities are $S^1\times \Sigma_i$3; then the spine is a disjoint union of solid tori and the construction reduces to an ordinary open book (Lisi et al., 2018, Min et al., 2024). This suggests that the essential difference is not the paper region but the replacement of the binding by a fibred “thickened surface.”

2. Contact structures supported by spinal open books

Spinal open books support contact structures by a direct analogue of Giroux compatibility. A contact form $S^1\times \Sigma_i$4 is a Giroux form for a spinal open book if $S^1\times \Sigma_i$5 is positive on the interior of each page and the Reeb vector field is positively tangent to the $S^1\times \Sigma_i$6-fibers of the spine (Min et al., 2024). In the formulation with boundary, one also requires compatibility of the characteristic foliation on boundary tori with the preferred meridians and paper fibration (Lisi et al., 2018).

A fundamental result is that, for a spinal open book admitting a smooth overlap, the space of Giroux forms is nonempty and contractible; consequently, the decomposition determines a canonical isotopy class of positive cooriented contact structures (Lisi et al., 2018). This is the spinal analogue of the uniqueness statement in Giroux theory. In the language used in filling results, one therefore speaks of a contact manifold $S^1\times \Sigma_i$7 supported by a spinal open book.

Several refinements organize the topology of the decomposition. A spinal open book is symmetric if all pages are diffeomorphic and each page meets each vertebra $S^1\times \Sigma_i$8 in the same number $S^1\times \Sigma_i$9 of boundary components. It is uniform if there exists a compact oriented surface $\Sigma_i$0 such that the paper components correspond to boundary components of $\Sigma_i$1, and each vertebra $\Sigma_i$2 carries a degree-$\Sigma_i$3 branched covering

$\Sigma_i$4

whose boundary restrictions encode the multiplicities on the adjacent paper boundary components (Min et al., 2024). A uniform spinal open book is Lefschetz-amenable if all such covering maps are unbranched (Min et al., 2024).

These refinements are important because they translate the geometry of the boundary contact manifold into branched-covering data over a single base surface. A plausible implication is that spinal open books are not merely a broader class of decompositions, but a bookkeeping device for the global topology that ordinary open books suppress.

3. Relation to Lefschetz fibrations and symplectic fillings

Spinal open books arise naturally as the boundary structures of bordered Lefschetz fibrations. If $\Sigma_i$5 is a bordered Lefschetz fibration over a compact oriented surface with nonempty boundary, then the horizontal boundary $\Sigma_i$6 becomes the spine and the vertical boundary $\Sigma_i$7 becomes the paper, producing a spinal open book on $\Sigma_i$8 (Lisi et al., 2018, Min et al., 2024). This is the reason the literature describes spinal open books as “the right contact boundary for Lefschetz fibrations over non disk bases” (Kaloti, 2013).

The geometric theory developed for bordered Lefschetz fibrations includes existence and contractibility of supported weakly convex, strongly convex, Liouville, and—when the fibration is allowable—almost Stein structures (Lisi et al., 2018). After smoothing corners, these give weak, strong, exact, or Stein fillings of the supported contact manifold (Lisi et al., 2018). In this sense, spinal open books mediate between contact boundary data and symplectic or Stein structures on a filling.

A major surgery construction in this setting is spine removal surgery, which removes selected spine components $\Sigma_i$9 and caps the adjacent page boundary components by disks, producing a symplectic cobordism

$M$0

with strongly concave boundary the original contact manifold and weakly convex boundary a new manifold carrying a generalized spinal open book (Lisi et al., 2018). This construction generalizes earlier cap constructions due to Eliashberg, Gay–Stipsicz, and Wendl (Lisi et al., 2018).

One major application is to partially planar domains and planar torsion. The geometric theory shows that if a contact 3-manifold has planar torsion, then it is not strongly symplectically fillable; under suitable cohomological conditions, it is not weakly fillable either (Lisi et al., 2018). This places spinal open books at the center of several fillability obstructions, not only filling classifications.

4. Planar spinal open books and finiteness phenomena

A spinal open book is called planar when its pages have genus zero (Min et al., 2024). In the notation of filling papers, this means that the page is a surface $M$1, or in the classical disk-with-holes notation $M$2 (Kaloti, 2013). Planarity is the key hypothesis behind the strongest finiteness theorems.

A foundational extension of Wendl’s planar-open-book theorem states that if a spinal open book $M$3 has a planar component of $M$4, then any symplectic filling of the supported contact manifold admits a Lefschetz fibration whose boundary is exactly that spinal open book (Kaloti, 2013). In the connected case used for geography results, if $M$5 is supported by a spinal open book with connected planar pages, then the geography set

$M$6

is finite; equivalently, there exists $M$7 such that every Stein filling satisfies

$M$8

(Kaloti, 2013). This extends the corresponding finiteness result for ordinary planar open books.

The proof strategy is specific to the spinal setting. Fillings are converted into Lefschetz fibrations over surfaces $M$9 rather than only over $M = M_\Sigma \cup M_P,$0, with monodromy written as

$M = M_\Sigma \cup M_P,$1

where the $M = M_\Sigma \cup M_P,$2 are positive Dehn twists about vanishing cycles and the commutators encode the base-surface monodromy (Kaloti, 2013). Because the fiber is planar, the mapping class group has a tractable abelianization, and multiplicity and joint multiplicity invariants can be used to bound the number of Dehn twists in any positive factorization, hence to bound $M = M_\Sigma \cup M_P,$3 (Kaloti, 2013).

The signature bound is obtained differently from the ordinary planar-open-book case. For contact manifolds supported by a spinal open book with connected spine $M = M_\Sigma \cup M_P,$4 and connected planar pages $M = M_\Sigma \cup M_P,$5, any two Stein fillings satisfy

$M = M_\Sigma \cup M_P,$6

so $M = M_\Sigma \cup M_P,$7 is an invariant of the contact boundary (Kaloti, 2013). Combined with bounded Euler characteristic, this yields bounded signature.

Later work significantly sharpened the picture. For a contact 3-manifold supported by a planar spinal open book, any minimal strong filling is symplectic deformation equivalent to the complement of a neighborhood of positive multisections in a bordered Lefschetz fibration (Min et al., 2024). This generalizes Wendl’s theorem from ordinary planar open books to spinal ones and shows that planar spinal fillings need not be ordinary PALFs: they may appear as complements of multisections, with additional singular phenomena at infinity (Min et al., 2024).

5. Exotic fibers, spinal mapping class groups, and monodromy factorizations

The modern monodromy theory of spinal open books enlarges the ordinary mapping class group framework. For a page $M = M_\Sigma \cup M_P,$8, the spinal mapping class group $M = M_\Sigma \cup M_P,$9 is generated by the subgroup $M_\Sigma$0 of mapping classes supported in the interior together with boundary interchanges $M_\Sigma$1, defined by properly embedded arcs $M_\Sigma$2 joining distinct boundary components (Min et al., 2024). Geometrically, a boundary interchange swaps two boundary components by a half-twist along $M_\Sigma$3, together with compensating rotations of the boundary circles (Min et al., 2024).

This enlargement is forced by the appearance of exotic fibers in fillings of planar spinal open books. In the pseudoholomorphic foliation picture, besides regular fibers and nodal Lefschetz fibers, one encounters embedded curves with one end asymptotic to a double cover of a Reeb orbit (Min et al., 2024). A local holomorphic model near such an exotic fiber is given by

$M_\Sigma$4

viewed on a domain in $M_\Sigma$5 (Min et al., 2024). In the compact picture, fibers near the exotic value change topology from a pair of pants to an annulus, and the monodromy around the exotic point is exactly a boundary interchange $M_\Sigma$6 (Min et al., 2024).

This leads to the notion of a positive admissible factorization of the page monodromy. For a uniform planar spinal open book with base $M_\Sigma$7 and an underlying representation

$M_\Sigma$8

one factors

$M_\Sigma$9

as a product of positive Dehn twists about homologically essential curves and boundary interchanges $\Sigma$0, with the number of boundary interchanges over a vertebra equal to the number of branch points of the corresponding covering $\Sigma$1 (Min et al., 2024). Fillings are then classified, up to symplectic deformation, by such factorizations modulo generalized Hurwitz equivalence (Min et al., 2024).

An application is the classification of strong fillings of parabolic torus bundles. For a rotational contact structure $\Sigma$2 on a torus bundle $\Sigma$3, the manifold $\Sigma$4 is supported by a planar spinal open book whose page is a punctured sphere $\Sigma$5 and whose spine is the disjoint union of a single annulus and $\Sigma$6 disks (Min et al., 2024). In particular, the torus bundle $\Sigma$7 is strongly fillable if and only if $\Sigma$8, and for $\Sigma$9 it admits a unique Stein filling up to symplectic deformation (Min et al., 2024). One striking consequence is that for annulus page $M_P$0 and annulus vertebra $M_P$1, the set $M_P$2 of monodromies giving Stein fillable spinal open books is not closed under multiplication, so it is not a monoid (Min et al., 2024). This sharply contrasts with the ordinary Giroux-open-book setting.

6. Higher-dimensional spinal open books and infinite fillings

Spinal open books also admit higher-dimensional analogues in which the page is replaced by a Liouville or Weinstein domain $M_P$3, and the spine is associated to a surface with boundary such as $M_P$4 (Zhou, 2023). In the simplest model, the contact manifold

$M_P$5

carries a contact structure by a Thurston–Winkelnkemper-type construction for spinal open books (Zhou, 2023). Zhou restricts to $M_P$6 and $M_P$7, so the contact manifold becomes the boundary

$M_P$8

of a product Weinstein domain (Zhou, 2023).

The key filling mechanism is representation-theoretic rather than factorization-theoretic. If $M_P$9 is a connected Riemann surface with boundary and $S^1$0 is a Weinstein domain, then any representation

$S^1$1

that sends the boundary class to $S^1$2 gives rise to a Weinstein filling of $S^1$3, diffeomorphic to the $S^1$4-bundle over $S^1$5 defined by that representation (Zhou, 2023). Specializing to $S^1$6, one varies a representation by sending one generator to a chosen compactly supported symplectomorphism $S^1$7 and the other to $S^1$8, obtaining a family of fillings $S^1$9 of the fixed contact boundary $\coprod S^1\times D^2$00 (Zhou, 2023).

With $\coprod S^1\times D^2$01 chosen as the plumbing of two copies of $\coprod S^1\times D^2$02 along three points, powers of Dehn–Seidel twists give infinitely many pairwise distinct Weinstein fillings in every odd dimension $\coprod S^1\times D^2$03, including the previously missing dimensions $\coprod S^1\times D^2$04 (Zhou, 2023). The distinction is detected by torsion in the homology of the total spaces, computed from exact sequences such as

$\coprod S^1\times D^2$05

and their analogues for bundles over $\coprod S^1\times D^2$06 (Zhou, 2023). This construction explicitly avoids the need to understand positive factorizations in the higher-dimensional symplectic mapping class group (Zhou, 2023).

From the perspective of geography, this higher-dimensional result complements the planar 3-dimensional finiteness theorems. The 2013 filling paper already emphasized that Baykur and Van Horn-Morris constructed counterexamples to Stipsicz’s conjecture using Lefschetz fibrations with nonclosed fiber of genus $\coprod S^1\times D^2$07 and nonclosed base of genus $\coprod S^1\times D^2$08, whose boundaries are described by spinal open books rather than ordinary open books (Kaloti, 2013). This suggests that planarity of the page, not merely the presence of an open-book-like decomposition, is the decisive finiteness condition.

7. Related frameworks and broader perspective

Several adjacent frameworks illuminate spinal open books by analogy or extension. Morse foliated open books encode contact manifolds with boundary by a circle-valued Morse function whose critical points lie on the boundary, producing pages whose topology changes by handle attachment or cutting along arcs; this setup yields a first-return monodromy $\coprod S^1\times D^2$09 on a subsurface and an extension of right-veering theory to the relative setting (Licata et al., 2021). The paper explicitly notes that these ideas align with spinal open books conceptually, even though spinal open books are not the primary object there (Licata et al., 2021).

Classical surgery and stabilization theory for ordinary contact open books also serves as a template. In higher dimensions, subcritical surgery along isotropic spheres in the binding corresponds to Weinstein handle attachment to the page, while Legendrian surgery along spheres lying in pages corresponds to composing the monodromy with a right-handed Dehn twist (Koert, 2010). Positive stabilization is realized by a handle attachment followed by such a Dehn twist and leaves the contact manifold unchanged (Koert, 2010). Although spinal open books are not treated explicitly in that work, it suggests how spinal stabilization moves should interact with Weinstein handle calculus.

Likewise, the algorithmic correspondence between classical supporting open books and contact surgery diagrams, together with ribbon moves implementing mapping class relations, does not mention spinal open books directly but is presented as conceptually adaptable to the spinal setting (Avdek, 2011). This suggests that a future “spinal Kirby calculus” would likely combine local pagewise ribbon constructions with additional gluing data along interface tori.

Finally, the notion of real open books on real 3-manifolds imposes equivariance of the fibration with respect to an involution and leads to a conjectural real Giroux correspondence (Ozturk et al., 2012). Although not formulated for spinal open books, the same principle—equivariance of the fibration, symmetry of monodromy, and equivariant stabilization—appears structurally compatible with a real version of spinal open book theory.

Taken together, these developments position spinal open books as a unifying boundary formalism in contact and symplectic topology. They generalize ordinary open books by replacing the binding with a fibred spine, encode the contact boundaries of Lefschetz fibrations over arbitrary bordered bases, provide a natural language for both filling finiteness and filling proliferation, and introduce genuinely new monodromy phenomena—especially exotic fibers and boundary interchanges—that do not arise in the classical planar-open-book setting (Kaloti, 2013, Lisi et al., 2018, Min et al., 2024, Zhou, 2023).