Nearly Lefschetz Fibrations

• Nearly Lefschetz fibrations are controlled generalizations of classical Lefschetz fibrations, allowing additional singular behavior such as connected round folds.
• They enhance monodromy and section theory by integrating ordinary Lefschetz points with exotic fibers, overcoming classical constraints in near-symplectic 4-manifolds.
• Recent studies link explicit genus‑1 normal forms, spin structure criteria, and boundary decompositions to classify these fibrations and their symplectic fillings.

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefschetz fibrations in order to describe near-symplectic 4-manifolds (Hayano, 2010). In the literature considered here, nearly Lefschetz fibrations are controlled relaxations of genuine Lefschetz fibrations in which one allows a prescribed additional singular behavior beyond isolated ordinary Lefschetz points. For closed 4-manifolds this appears as the simplified broken Lefschetz fibration, with isolated Lefschetz singularities together with a connected circle of indefinite-fold, or round, singularities (Hayano, 2011). In recent symplectic filling theory, a nearly Lefschetz fibration over the disk is a smooth map on a compact 4-manifold with corners whose singularities consist of ordinary Lefschetz points and finitely many exotic fibers locally modeled on $\{y^2=x\}$, producing spinal rather than classical open books on the boundary (Plamenevskaya et al., 28 Jul 2025). Across these settings, the extra singularity enlarges the available monodromy, section theory, and filling theory beyond the classical Lefschetz regime.

1. Definitions and local models

A broken Lefschetz fibration $f\colon M\to B$, where $M$ is an oriented 4-manifold and $B$ an oriented surface, is a smooth map whose only critical points are Lefschetz singularities and indefinite folds. In compatible complex local coordinates $(z_1,z_2)$ on $M$ and $\xi$ on $B$, a Lefschetz singularity is modeled by

$f(z_1,z_2)=\xi=z_1z_2,$

while in real coordinates $(t,x_1,x_2,x_3)$ on $f\colon M\to B$0 and $f\colon M\to B$1 on $f\colon M\to B$2, an indefinite fold is modeled by

$f\colon M\to B$3

The isolated Lefschetz points are denoted $f\colon M\to B$4, and the indefinite folds form a connected 1-manifold $f\colon M\to B$5 (Hayano, 2011).

A broken Lefschetz fibration $f\colon M\to B$6 is called simplified when three conditions hold: $f\colon M\to B$7 is injective, $f\colon M\to B$8 is a single embedded circle and all fibers are connected, and all Lefschetz critical points lie on the higher-genus side. Under these hypotheses, $f\colon M\to B$9 is the disjoint union of two disks $M$0, the fibers over them differ in genus by one, and the region over a tubular neighborhood of $M$1 is the round cobordism (Hayano, 2011). In the genus-1 case, the higher side has genus $M$2, the lower side genus $M$3, and the round cobordism can be described handle-theoretically as an untwisted or twisted round 1-handle, equivalently a pair of ordinary 1- and 2-handles attached with algebraic intersection zero or two on the belt sphere (Hayano, 2010).

The recent disk-based notion is formulated differently. Let $M$4 be a compact 4-manifold with corners and boundary decomposition

$M$5

A nearly Lefschetz fibration over $M$6 is a smooth map $M$7 such that over the interior of $M$8 it is a fibration with connected fiber $M$9, its restriction to $B$0 is exactly this surface bundle, it has finitely many ordinary Lefschetz critical points locally modeled on $B$1, and finitely many exotic fibers locally modeled on the branched-cover curve $B$2, equivalently by

$B$3

Away from the exotic fibers, $B$4 carries $B$5 to $B$6 as a smooth bundle (Plamenevskaya et al., 28 Jul 2025).

These two formalisms are not identical, but both replace the rigidity of genuine Lefschetz fibrations by one additional singular mechanism. This suggests that “nearly Lefschetz” is best understood as a controlled departure from the honest Lefschetz category rather than as a single universally fixed local model.

2. Monodromy and boundary structures

For a simplified broken Lefschetz fibration, the higher-side restriction

$B$7

is an honest genus-$B$8 Lefschetz fibration with vanishing cycles $B$9 and Hurwitz sequence

$(z_1,z_2)$0

If $(z_1,z_2)$1 is the vanishing cycle of the indefinite fold, then Baykur’s analysis shows that the total monodromy around $(z_1,z_2)$2 preserves $(z_1,z_2)$3 and lies in the kernel of the cut-along-$(z_1,z_2)$4 map

$(z_1,z_2)$5

Conversely, any factorization $(z_1,z_2)$6 gives a simplified broken Lefschetz fibration with indefinite-fold vanishing cycle $(z_1,z_2)$7 (Hayano, 2011). The round cobordism is built by first attaching a fiberwise 2-handle along the nonseparating curve $(z_1,z_2)$8 with fiber framing and then a 3-handle, so the monodromy condition and the handle picture are tightly linked.

For nearly Lefschetz fibrations over $(z_1,z_2)$9, the monodromy contains two kinds of generators. Ordinary critical values determine vanishing cycles, that is, embedded closed curves in the fiber about which the monodromy is a positive Dehn twist. Exotic fibers determine vanishing arcs, along which the monodromy is a positive boundary-interchange, namely a half-twist swapping two boundary components of the page. As a result, the monodromy lives not in the ordinary mapping class group alone but in the spinal mapping class group generated by Dehn twists and boundary-interchanges (Plamenevskaya et al., 28 Jul 2025).

This distinction is reflected on the boundary. A standard Lefschetz fibration over $M$0 induces a classical open book, in which each binding component appears once in the boundary of each page. A nearly Lefschetz fibration induces instead a spinal open book: the 3-manifold decomposes as

$M$1

where $M$2 is a bundle with page $M$3, $M$4 is a union of solid tori, and each page can meet a torus in multiple longitudes. The multiplicities of these longitudes are part of the boundary data (Plamenevskaya et al., 28 Jul 2025). In this sense, the passage from standard to nearly Lefschetz fibrations is simultaneously a change in local singularity theory and in the global boundary category.

3. Sections and the failure of classical Lefschetz constraints

For relatively minimal genuine Lefschetz fibrations $M$5 of genus $M$6, classical constraints are stringent. Smith proved that there are only finitely many homotopy classes of sections $M$7, and Smith and Stipsicz independently proved that every section satisfies

$M$8

unless the fibration is the trivial product $M$9 (Hayano, 2011).

Hayano showed that neither phenomenon survives in the simplified broken setting. For every $\xi$0, there exists a nontrivial genus-$\xi$1 simplified broken Lefschetz fibration

$\xi$2

with no $\xi$3-sphere in any fiber and with infinitely many pairwise non-homotopic sections $\xi$4 (Hayano, 2011). The construction chooses a simple closed curve $\xi$5, takes the Hurwitz system $\xi$6, and selects the indefinite-fold cycle $\xi$7 so that $\xi$8. The resulting Kirby diagram presents

$\xi$9

The sections $B$0 are obtained by trivializing over the lower-genus side the boundary torus of $B$1 with winding number $B$2 around one fundamental loop in $B$3. A computation of $B$4 then shows that these section classes are all distinct.

Hayano also proved that for any integer $B$5 and any $B$6 there exists a nontrivial genus-$B$7 simplified broken Lefschetz fibration admitting a section $B$8 with

$B$9

(Hayano, 2011). The monodromy-based construction works on $f(z_1,z_2)=\xi=z_1z_2,$0, choosing a nonseparating curve $f(z_1,z_2)=\xi=z_1z_2,$1 and curves $f(z_1,z_2)=\xi=z_1z_2,$2 so that

$f(z_1,z_2)=\xi=z_1z_2,$3

where $f(z_1,z_2)=\xi=z_1z_2,$4 is the boundary circle. Kas’s handle-attachment construction gives a Lefschetz fibration over $f(z_1,z_2)=\xi=z_1z_2,$5, after which one attaches the round 2-handle along $f(z_1,z_2)=\xi=z_1z_2,$6 and caps with a trivial $f(z_1,z_2)=\xi=z_1z_2,$7 piece to obtain a closed simplified broken Lefschetz fibration over $f(z_1,z_2)=\xi=z_1z_2,$8. The section is then read off from the handle picture.

These results indicate that the finiteness of section homotopy classes and the negativity of section square are not features of “broken” or “nearly” Lefschetz theory in general; they are genuinely classical Lefschetz phenomena, destroyed by the presence of the round singularity.

4. Spin structures and obstruction theory

Let $f(z_1,z_2)=\xi=z_1z_2,$9 be a genus-$(t,x_1,x_2,x_3)$0 simplified broken Lefschetz fibration with Lefschetz vanishing cycles $(t,x_1,x_2,x_3)$1 and round-vanishing cycle $(t,x_1,x_2,x_3)$2. Suppose $(t,x_1,x_2,x_3)$3 admits a dual class $(t,x_1,x_2,x_3)$4 to a lower-side fiber $(t,x_1,x_2,x_3)$5, meaning

$(t,x_1,x_2,x_3)$6

This hypothesis holds, for example, whenever the higher-plus-round region is simply connected, or if the fibration has a section (Hayano, 2011).

Hayano’s spin criterion states that $(t,x_1,x_2,x_3)$7 admits a spin structure if and only if two conditions hold. First, there exists a $(t,x_1,x_2,x_3)$8-quadratic form

$(t,x_1,x_2,x_3)$9

with respect to the intersection pairing on $f\colon M\to B$00 such that

$f\colon M\to B$01

Second, the self-intersection $f\colon M\to B$02 is even (Hayano, 2011).

Equivalently, the monodromy data determine the relevant contribution to the second Stiefel-Whitney class $f\colon M\to B$03 through the quadratic form $f\colon M\to B$04, while the only additional global obstruction is the parity of the dual class $f\colon M\to B$05. In the genuine Lefschetz case, where there is no round singularity, the condition reduces to the usual Stipsicz-Gompf criterion $f\colon M\to B$06. The extra requirement $f\colon M\to B$07 isolates precisely how the round vanishing cycle enters the obstruction theory.

This criterion is structurally significant because it shows that the broken setting retains a mapping-class-group formulation of spin obstructions, but only after incorporating data from the non-Lefschetz singularity. The obstruction theory therefore parallels the classical one without coinciding with it.

5. Genus-$f\colon M\to B$08 normal forms and low-complexity classification

The genus-$f\colon M\to B$09 case admits a particularly explicit description because the mapping class group $f\colon M\to B$10 is identified with $f\colon M\to B$11. Writing

$f\colon M\to B$12

a genus-$f\colon M\to B$13 simplified broken Lefschetz fibration with $f\colon M\to B$14 Lefschetz critical points has a Hurwitz system

$f\colon M\to B$15

where each $f\colon M\to B$16 is a right-handed Dehn twist $f\colon M\to B$17 along a slope $f\colon M\to B$18, and the product

$f\colon M\to B$19

is the boundary monodromy of the higher-side torus bundle (Hayano, 2010). Because that torus bundle has a section, the monodromy fixes one slope, hence

$f\colon M\to B$20

for some integer $f\colon M\to B$21.

Hayano proved that after simultaneous conjugations and elementary Hurwitz moves, the higher-side monodromy can be put in the normal form

$f\colon M\to B$22

that is, a block

$f\colon M\to B$23

followed by a twisted block

$f\colon M\to B$24

(Hayano, 2010). For $f\colon M\to B$25, the allowable twisted blocks are exactly the standard sequences

$f\colon M\to B$26

$f\colon M\to B$27

and for $f\colon M\to B$28,

$f\colon M\to B$29

with

$f\colon M\to B$30

This leads to a complete classification for the low-complexity range $f\colon M\to B$31. When $f\colon M\to B$32, only $f\colon M\to B$33 occurs. When $f\colon M\to B$34, the possibilities are $f\colon M\to B$35 or $f\colon M\to B$36. When $f\colon M\to B$37, the possibilities are $f\colon M\to B$38, $f\colon M\to B$39, or $f\colon M\to B$40; in the purely Lefschetz limit, $f\colon M\to B$41 connects to the elliptic-surface picture, and when the round handle is absent the standard 12$f\colon M\to B$42-term factorization recovers the elliptic surfaces $f\colon M\to B$43 (Hayano, 2010).

The diffeomorphism types of the total spaces are obtained by Kirby calculus. The families $f\colon M\to B$44 realize

$f\colon M\to B$45

while the families $f\colon M\to B$46 realize

$f\colon M\to B$47

where $f\colon M\to B$48 or $f\colon M\to B$49 (Hayano, 2010). The numerical invariants are correspondingly explicit: $f\colon M\to B$50 since the round handle contributes zero to $f\colon M\to B$51, and in the pure $f\colon M\to B$52 case the signature is $f\colon M\to B$53. Hayano further conjectured that the families $f\colon M\to B$54 and $f\colon M\to B$55, together with their possible $f\colon M\to B$56 summands, exhaust all genus-$f\colon M\to B$57 simplified broken Lefschetz fibrations with non-empty round locus (Hayano, 2010).

6. Spinal open books, multisections, and fillings of sandwiched links

In the symplectic-topological setting of sandwiched singularities, nearly Lefschetz fibrations over the disk are tied to spinal open books and filling classification. A spinal open book on a 3-manifold $f\colon M\to B$58 is a decomposition

$f\colon M\to B$59

where $f\colon M\to B$60 is a bundle with compact surface pages $f\colon M\to B$61 with boundary, and $f\colon M\to B$62 is the spine, glued so that each page meets each torus in a collection of longitudes, possibly more than one. When the spine is a union of solid tori and the pages are planar, the open book is called planar uniform over $f\colon M\to B$63, and each binding component carries a multiplicity $f\colon M\to B$64 equal to the number of longitudes contributed by a page (Plamenevskaya et al., 28 Jul 2025).

Roy, Min, and Wang, building on work of Lisi, Van Horn-Morris, and Wendl, proved that if $f\colon M\to B$65 is supported by a planar spinal open book uniform over $f\colon M\to B$66, then any minimal strong or Stein filling $f\colon M\to B$67 admits the structure of a positive allowable nearly Lefschetz fibration inducing that spinal open book on $f\colon M\to B$68; moreover, the number of exotic fibers is determined solely by the binding multiplicities (Plamenevskaya et al., 28 Jul 2025). This theorem supplies the passage from contact boundary data to nearly Lefschetz structures.

The corresponding filling constructions are encoded by immersed disk arrangements

$f\colon M\to B$69

subject to three conditions: each $f\colon M\to B$70 is a simple branched cover; all self-intersections and mutual intersections are positive transverse multipoints, with every branch point or intersection marked; and the total number of marked points on $f\colon M\to B$71 equals a prescribed weight $f\colon M\to B$72 (Plamenevskaya et al., 28 Jul 2025). After blowing up the $f\colon M\to B$73 marked points, the strict transforms $f\colon M\to B$74 become a multisection of the blown-up trivial fibration, and the complement

$f\colon M\to B$75

carries a natural positive allowable nearly Lefschetz fibration. Ordinary singular fibers come from blowing down intersections, and exotic fibers come from branch points. The monodromy around an ordinary critical value is a Dehn twist enclosing exactly the holes corresponding to the disks meeting there, while the monodromy around an exotic value is a boundary-interchange of the two adjacent holes.

This framework yields two central existence statements. First, if $f\colon M\to B$76 is compatible with the decorated germ $f\colon M\to B$77 of a sandwiched singularity, then $f\colon M\to B$78 admits a Stein structure filling the canonical contact link. Second, every minimal strong symplectic filling of the contact link of a sandwiched singularity arises, up to deformation, as some $f\colon M\to B$79, equivalently via a positive allowable nearly Lefschetz fibration whose vanishing cycles and arcs can be packaged into a braided wiring diagram, and conversely a wiring diagram produces an immersed-disk arrangement (Plamenevskaya et al., 28 Jul 2025).

The same formalism detects unexpected Stein fillings, namely Stein fillings not diffeomorphic, indeed not homeomorphic, to any Milnor fiber of any analytic smoothing. The incidence matrix

$f\colon M\to B$80

records how many marked points each $f\colon M\to B$81 carries. Akhmedov, Ozbagci, and Popescu-Pampu show topologically that any two fillings with the same boundary marking have the same matrix, while de Jong–van Straten’s algebraic picture deformations impose additional constraints on realizable matrices. For the decorated germ consisting of $f\colon M\to B$82 lines through the origin in $f\colon M\to B$83, each weighted $f\colon M\to B$84, one obtains at least $f\colon M\to B$85 pairwise non-homeomorphic Stein fillings by choosing DJVS arrangements with incidence matrices not realizable by any complex picture deformation; each such filling carries its unique positive allowable nearly Lefschetz fibration, with monodromy factoring as a product of boundary-interchanges followed by Dehn twists (Plamenevskaya et al., 28 Jul 2025).

In this symplectic-filling context, nearly Lefschetz fibrations generalize planar Lefschetz fibrations to the spinal setting, arise naturally from complements of multisections, and parametrize all minimal symplectic fillings of sandwiched contact links, including Stein fillings outside the classical algebraic Milnor-fiber picture.