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Jeremy Van Horn-Morris

Publications and source records attributed to Jeremy Van Horn-Morris.

18 recordsLinked to original sources
arXiv2026

On symplectic fillings of spinal open book decompositions I: Geometric constructions

A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e.g. on the boundary of a symplectic filling with a Lefschetz fibration over any compact oriented surface with boundary. In this first paper of a two-part series, we introduce the basic notions relating spinal open books to contact structures and symplectic or Stein structures on Lefschetz fibrations, leading to the definition of a new symplectic cobordism construction called spine removal surgery, which generalizes previous constructions due to Eliashberg, Gay-Stipsicz and the third author. As an application, spine removal yields a large class of new examples of contact manifolds that are not strongly (and sometimes not weakly) symplectically fillable. This paper also lays the geometric groundwork for a theorem to be proved in part II, where holomorphic curves are used to classify the symplectic and Stein fillings of contact 3-manifolds admitting a spinal open book with a planar page.

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arXiv2026

On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic and Stein fillings can be classified up to deformation equivalence in terms of diffeomorphism classes of Lefschetz fibrations. This extends previous results of the third author to a much wider class of contact manifolds, which we illustrate here by classifying the strong and Stein fillings of all oriented circle bundles with non-tangential $S^1$-invariant contact structures. Further results include new vanishing criteria for the ECH contact invariant and algebraic torsion in SFT, classification of fillings for certain non-orientable circle bundles, and a general "symplectic quasiflexibility" result about deformation classes of Stein structures in real dimension four.

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arXiv2025

Computing Heegaard Floer invariants of closed contact 3-manifolds from open books

We present two SageMath programs that build on and improve upon Sucharit Sarkar's hf-hat. Given an abstract open book and a collection of pairwise disjoint properly embedded arcs on a page of the open book, the first program, hf-hat-obd, can be used to analyze the resulting Heegaard diagram, while the second, hf-hat-obd-nice computes the hat version of Heegaard Floer homology of the closed oriented 3-manifold described by the Heegaard diagram as long as the latter is nice. We also provide an auxiliary program, makenice, that can be used to produce a nice Heegaard diagram out of any abstract open book and a collection of pairwise disjoint properly embedded arcs on a page of the open book. The primary applications of hf-hat-obd-nice are to the computation of the Ozsváth--Szabó contact invariant and to the detection of finiteness of spectral order, which is a Stein fillability obstruction that is stronger than the vanishing of the Ozsváth--Szabó contact invariant.

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arXiv2023

Splitting Algebraic Singular Fibrations via Perturbation of Branch Covers

In a previous paper \cite{SV}, the authors studied the isolated singular fibers that can occur in algebraic fibrations of certain genus two fibrations. There the goal was to determine their monodromy factorizations with the goal of determining a dictionary between a set of curve configurations and certain words in the mapping class group. Each such curve configuration was originally cataloged by Namikawa and Ueno \cite{NamikawaUeno-list} in their list of genus two fibrations. We studied four families of polynomials, we restricted to fibrations whose fibers have boundary, considering the isolated affine singularity referenced in \cite{NamikawaUeno-list}. We resolved the singularities and, using carefully chosen perturbations, deformed them into Lefschetz fibrations and determined their monodromy factorizations. In two of those families, we also obtained strong information about how the central fiber compactifies in a fibration with closed fibers. In this paper we work on the other two cases by recreating the singular fibers using a different family of polynomials. In \cite{SV}, all the algebraic curves in the fibrations were given expressly as hyperelliptic equations. We utilized this symmetry both to construct the deformation and to recover the monodromy factorization. In this paper the curves are no longer expressly hyperelliptic--the quotient curve is no longer just $\CC$, the branch curves are no longer embedded in $\CC^2$, and the fibrations utilized in the quotient are now more complicated. We do, though, recover the desired behavior of the compactification of the central fiber, its deformation to a Lefschetz fibration, and the corresponding monodromy factorization.

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arXiv2023

Singular fibers in algebraic fibrations of genus 2 and their monodromy factorizations

Kodaira's classification of singular fibers in elliptic fibrations and its translation into the language of monodromies and Lefschetz fibrations has been a boon to the study of 4-manifolds. In this article, we begin the work of translating between singular fibers of genus 2 families of algebraic curves and the positive Dehn twist factorizations of Lefschetz fibrations for a certain subset of the singularities described by Namikawa and Ueno in the 70s. We look at four families of hypersurface singularities in $\mathbb{C}^3$. Each hypersurface comes equipped with a fibration by genus 2 algebraic curves which degenerate into a single singular fiber. We determine the resolution of each of the singularities in the family and find a flat deformation of the resolution into simpler pieces, resulting in a fibration of Lefschetz type. We then record the description of the Lefschetz as a positive factorization in Dehn twists. This gives us a dictionary between configurations of curves and monodromy factorizations for some singularities of genus 2 fibrations.

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arXiv2019

Filtering the Heegaard Floer contact invariant

We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.

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arXiv2016

Fillings of genus-1 open books and 4-braids

We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz classes of quasipositive factorizations, yielding in particular an infinite family of knotted complex analytic annuli in the 4-ball bounding the same transverse link up to transverse isotopy. These realize the smallest possible examples in terms of the number of boundary components a genus-1 mapping class and the number of strands a braid can have with infinitely many positive/quasipositive factorizations.

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arXiv2016

Algebraic torsion via Heegaard Floer homology

We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.

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arXiv2015

Fillings of unit cotangent bundles

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For exact fillings, we show that the rational homology agrees with that of the disk cotangent bundle, and that the integral homology takes on finitely many possible values: for example, if $g-1$ is square-free, then any exact filling has the same integral homology and intersection form as $DT^*Σ_g$.

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arXiv2015

Positive factorizations of mapping classes

In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g,n, we tell whether or not there exist genus-g Lefschetz pencils with n base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. We observe that only symplectic 4-manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.

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arXiv2015

Monoids in the mapping class group

In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.

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arXiv2012

Topological complexity of symplectic 4-manifolds and Stein fillings

We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by Auroux, Smith and Wajnryb, and a more general variant of it raised by Korkmaz, Ozbagci and Stipsicz, independently.

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arXiv2012

Families of contact 3-manifolds with arbitrarily large Stein fillings

We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily big euler characteristics and arbitrarily small signatures ---which disproves a conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in [24].

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arXiv2011

Cabling, contact structures and mapping class monoids

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

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arXiv2009

Planar open books, monodromy factorizations, and symplectic fillings

We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fillable and non-fillable tight contact structures on certain Seifert fibered spaces.

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arXiv2009

Fibered Transverse Knots and the Bennequin Bound

We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold $(M,ξ)$ with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is $ξ.$ This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also note that this allows the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on $S^3$ we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knots types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular. we give conditions under which quasi-postitive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. In the new version we also prove that our main result can be used to show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support.

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