Heegaard Splittings Research Articles

Taut foliations of 3-manifolds with Heegaard genus 2

Taut foliations of 3-manifolds with Heegaard genus 2

Trisections obtained by trivially regluing surface-knots

Let S be a P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^2$$\\end{document}-knot which is the connected sum of a 2-knot with normal Euler number 0 and an unknotted P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P^2$$\\end{document}-knot with normal Euler number ±2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\pm }{2}$$\\end{document} in a closed 4-manifold X with trisection TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{X}$$\\end{document}. Then, we show that the trisection of X obtained by the trivial gluing of relative trisections of ν(S)¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{\ u (S)}$$\\end{document} and X-ν(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X-\ u (S)$$\\end{document} is diffeomorphic to a stabilization of TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{X}$$\\end{document}. It should be noted that this result is not obvious since boundary-stabilizations introduced by Kim and Miller are used to construct a relative trisection of X-ν(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X-\ u (S)$$\\end{document}. As a corollary, if X=S4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=S^4$$\\end{document} and TX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_X$$\\end{document} was the genus 0 trisection of S4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^4$$\\end{document}, the resulting trisection is diffeomorphic to a stabilization of the genus 0 trisection of S4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^4$$\\end{document}. This result is related to the conjecture that is a 4-dimensional analogue of Waldhausen’s theorem on Heegaard splittings.

Double Johnson filtrations for mapping class groups

In this paper, we first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group [Formula: see text] acting on another group [Formula: see text] equipped with a filtration indexed by a “good” ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid [Formula: see text] of pairs on non-negative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group [Formula: see text] of a surface [Formula: see text] with one boundary component, equipped with the normal subgroups [Formula: see text], [Formula: see text] of [Formula: see text] associated to a standard Heegaard splitting of the [Formula: see text]-sphere. We also consider the case where the group [Formula: see text] is the automorphism group of a free group.

On smooth functions with two critical values

We prove that every smooth closed connected manifold admits a smooth real-valued function with only two critical values such that the set of minima (or maxima) can be arbitrarily prescribed, as soon as this set is a finite subcomplex of the manifold (we call a function of this type a Reeb function). In analogy with Reeb’s Sphere Theorem, we use such functions to study the topology of the underlying manifold. In dimension 3, we give a characterization of manifolds having a Heegaard splitting of genus g in terms of the existence of certain Reeb functions. Similar results are proved in dimension n≥5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 5$$\\end{document}.

Reducing spheres of genus-2 Heegaard splitting of S3

The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, [Formula: see text], acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann [Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana 10 (2004) 503–514] uses this action to prove that [Formula: see text] is finitely generated. In this paper, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the [Formula: see text]. Using this we give an alternate proof of the finite generation of [Formula: see text] assuming the finite generation of the stabilizer of the standard reducing sphere.

Modular factorization of superconformal indices

Superconformal indices of four-dimensional mathcal{N} = 1 gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an SL(3, ℤ) and SL(2, ℤ) ⋉ ℤ2 transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic Γ function. This leads to our conjecture for the “modular factorization” of superconformal lens indices of general mathcal{N} = 1 gauge theories. We provide evidence for the conjecture in the context of the free chiral multiplet and SQED and sketch the extension of our arguments to more general gauge theories. Assuming the validity of the conjecture, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with SL(3, ℤ). Finally, we use this framework to propose a formula for the general lens space index.

An algorithmic method to compute plat slide moves in 3-manifolds of Heegaard genus two

This article is motivated by the equivalence problem of links embedded in 3-manifolds of Heegaard genus two. Casali and Grasselli introduced a way to encode every such 3-manifold M by an admissible 6-tuples of integers, that also determines a genus two Heegaard splitting of the manifold M. We construct an algorithm (implemented in C++) which, starting from an admissible 6-tuple of integers, allows to find the words in B2,2n, the braid group on 2n strands on a genus two surface, that realize the plat slide moves in that manifold. This extends the work of Cattabriga and Gabrovšek, who carried out this computation for 3-manifolds of Heegaard genus one. As an application of our algorithm, we compute the words that represent the plat slide moves for all orientable prime 3-manifolds of Heegaard genus two that can be triangulated with at most 42 colored tetrahedra.

Sutured instanton homology and Heegaard diagrams

Suppose $\mathcal {H}$ is an admissible Heegaard diagram for a balanced sutured manifold $(M,\gamma )$. We prove that the number of generators of the associated sutured Heegaard Floer complex is an upper bound on the dimension of the sutured instanton homology $\mathit {SHI}(M,\gamma )$. It follows, in particular, that strong L-spaces are instanton L-spaces.

On invariants of surfaces in the 3-sphere

In this paper we study isotopy classes of closed connected orientable surfaces in the standard [Formula: see text]-sphere. Such a surface splits the [Formula: see text]-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a [Formula: see text]-component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial [Formula: see text]-handles can recover the original surface. Therefore, we can reduce the study of surfaces in the [Formula: see text]-sphere to that of [Formula: see text]-component handlebody-links up to stabilizations. Then, by using [Formula: see text]-families of quandles, we construct invariants of [Formula: see text]-component handlebody-links up to attaching trivial [Formula: see text]-handles, which lead to invariants of surfaces in the [Formula: see text]-sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the [Formula: see text]-sphere.

Note on primitive disk complexes

Given a Heegaard splitting of the 3-sphere, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting, spanned by the vertices of primitive disks. It is an open question whether the primitive disk complex is connected or not when the genus of the splitting is greater than three. In this note, we prove that a quotient of the primitive disk complex, called the homotopy primitive disk complex, is connected.

Equivariant Heegaard genus of reducible 3-manifolds

AbstractThe equivariant Heegaard genus of a 3-manifold W with the action of a finite group G of diffeomorphisms is the smallest genus of an equivariant Heegaard splitting for W. Although a Heegaard splitting of a reducible manifold is reducible and although if W is reducible, there is an equivariant essential sphere, we show that equivariant Heegaard genus may be super-additive, additive, or sub-additive under equivariant connected sum. Using a thin position theory for 3-dimensional orbifolds, we establish sharp bounds on the equivariant Heegaard genus of reducible manifolds, similar to those known for tunnel number. Along the way, we make use of a new invariant for W which is much better behaved under equivariant sums.

Some computations on instanton knot homology

In a recent paper, the first author and his collaborator developed a method to compute an upper bound of the dimension of instanton Floer homology via Heegaard diagrams of [Formula: see text]-manifolds. In this paper, for a knot inside [Formula: see text], we further introduce an algorithm that computes an upper bound of the dimension of instanton knot homology from knot diagrams. We test the algorithm with all knots up to seven crossings as well as a more complicated knot [Formula: see text]. In the second half of the paper, we show that if the instanton knot Floer homology of a knot has a specified form, then the knot must be an instanton L-space knot.

On Stiefel’s parallelizability of 3-manifolds

We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.

Bordered Floer homology and contact structures

Abstract We introduce a contact invariant in the bordered sutured Heegaard Floer homology of a three-manifold with boundary. The input for the invariant is a contact manifold $(M, \xi , \mathcal {F})$ whose convex boundary is equipped with a signed singular foliation $\mathcal {F}$ closely related to the characteristic foliation. Such a manifold admits a family of foliated open book decompositions classified by a Giroux correspondence, as described in [LV20]. We use a special class of foliated open books to construct admissible bordered sutured Heegaard diagrams and identify well-defined classes $c_D$ and $c_A$ in the corresponding bordered sutured modules. Foliated open books exhibit user-friendly gluing behavior, and we show that the pairing on invariants induced by gluing compatible foliated open books recovers the Heegaard Floer contact invariant for closed contact manifolds. We also consider a natural map associated to forgetting the foliation $\mathcal {F}$ in favor of the dividing set and show that it maps the bordered sutured invariant to the contact invariant of a sutured manifold defined by Honda–Kazez–Matić.

Small knots of large Heegaard genus

Small knots of large Heegaard genus

4–Manifold invariants from Hopf algebras

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.

Classifying compact 4-manifolds via generalized regular genus and G-degree

$(d+1)$-colored graphs, i.e., edge-colored graphs that are $(d+1)$-regular, have already been proved to be a useful representation tool for compact PL $d$-manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role. The present paper focuses in particular on generalized regular genus and G-degree: the first one extending to higher dimension the classical notion of Heegaard genus for 3-manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two. We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums. We also compute both generalized regular genus and G-degree for interesting classes of compact $d$-manifolds, such as handlebodies, products of closed manifolds by the interval and $\mathbb D^2$-bundles over $\mathbb S^2.$ The main results of the paper concern dimension 4, where we obtain the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with G-degree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to G-degree 24.

Generalized torsion for hyperbolic 3‐manifold groups with arbitrary large rank

AbstractLet be a group and a non‐trivial element in . If some non‐empty finite product of conjugates of equals to the trivial element, then is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic 3‐manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer there are infinitely many closed hyperbolic 3‐manifolds which enjoy the property: (i) the Heegaard genus of is , (ii) the rank of is , and (ii) has a generalized torsion element. Furthermore, we may choose as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.

The finiteness conjecture for skein modules

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.

A model for random three-manifolds

We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component.We prove a lawof large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number. We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary.We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini–Schramm limit of our sequence of random manifolds.