Heegaard Splitting Of Genus Research Articles

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}.

Heegaard genus and complexity of fibered knots

We prove that if a fibered knot K $K$ with genus greater than 1 in a three-manifold M $M$ has a sufficiently complicated monodromy, then K $K$ induces a minimal genus Heegaard splitting P $P$ that is unique up to isotopy, and small genus Heegaard splittings of M $M$ are stabilizations of P $P$ . We provide a complexity bound in terms of the Heegaard genus of M $M$ . We also provide global complexity bounds for fibered knots in the three-sphere and lens spaces.

An equivalent description of the lens space L(p,q) with p prime

In the paper, we give an equivalent description of the lens space [Formula: see text] with [Formula: see text] prime in terms of any corresponding Heegaard diagrams as follows: Let [Formula: see text] be a closed orientable 3-manifold with [Formula: see text] and [Formula: see text] a Heegaard splitting of genus [Formula: see text] for [Formula: see text] with an associated Heegaard diagram [Formula: see text]. Assume [Formula: see text] is a prime integer. Then [Formula: see text] is homeomorphic to the lens space [Formula: see text] if and only if there exists an embedding [Formula: see text] such that [Formula: see text] bounds a complete system of surfaces for [Formula: see text].

On large orientation-reversing finite group-actions on 3-manifolds and equivariant Heegaard decompositions

We consider finite group-actions on closed, orientable and nonorientable 3-manifolds; such a finite group-action leaves invariant the two handlebodies of a Heegaard splitting of M of some genus g. The maximal possible order of a finite group-action of an orientable or nonorientable handlebody of genus $$g>1$$ is $$24(g-1)$$, and in the present paper we characterize the 3-manifolds M and groups G for which the maximal possible order $$|G| = 24(g-1)$$ is obtained, for some G-invariant Heegaard splitting of genus $$g>1$$. If M is reducible then it is obtained by doubling an action of maximal possible order $$24(g-1)$$ on a handlebody of genus g. If M is irreducible then it is a spherical, Euclidean or hyperbolic manifold obtained as a quotient of one of the three geometries by a normal subgroup of finite index of a Coxeter group associated to a Coxeter tetrahedron, or of a twisted version of such a Coxeter group.

The mapping class groups of reducible Heegaard splittings of genus two

The manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, $\mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-$2$ splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces, and show that the mapping class group of the genus-$2$ Heegaard splitting is finitely presented for any lens space by giving its explicit presentation. As an application, we show that the fundamental groups of the spaces of the genus-$2$ Heegaard splittings of lens spaces are all finitely presented.

A topologically minimal canonical Heegaard splitting of a surface bundle is critical

Every surface bundle with genus [Formula: see text] fiber has a canonical Heegaard splitting of genus [Formula: see text]. In this paper, we discuss the topological indices of such Heegaard surfaces and prove the canonical Heegaard splitting of a surface bundle is topologically minimal if and only if it is critical, that is, its topological index is 2.

On the group action of Mod(M,F) on the disk complex

Let (V,W;F) be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible 3-manifold M. Then Mod(M,F) naturally acts on the disk complex D(F) as a group action. In this article, we prove if F is topologically minimal and its topological index is two, then the orbit of any element of D(F) for this group action consists of infinitely many elements. Moreover, we prove there are at most two elements of D(F) whose orbits are finite if the genus of F is three.

Stabilizing Heegaard splittings of high-distance knots

Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminus\eta(K)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Further, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to these Heegaard surfaces.

Views of Real Projective 3-Space by Stable Maps into the Plane

ABSTRACTSmooth stable maps into the plane enable us to get the graphical views of the source manifolds. In this article, we present two such maps of the real projective 3-space constructed from typical two projections of a tetrahedron to the plane. Tri- and quarto-sections of the real projective 3-space and Heegaard splittings of genus four and five can be obtained naturally using these maps. A question is posed on views of projective n-spaces of n ⩾ 4.

A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical

Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M. In this article, we prove that either the disk complex D(F) is contractible or F is critical. Hence, the topological index of F is two if F is topologically minimal.

The Heegaard distances cover all nonnegative integers

In this paper, we prove that (1) For any integers $n\geq 1$ and $g\geq 2$, there is a closed 3-manifold $M_{g}^{n}$ which admits a distance $n$ Heegaard splitting of genus $g$ except that the pair of $(g, n)$ is $(2, 1)$. Furthermore, $M_{g}^{n}$ can be chosen to be hyperbolic except that the pair of $(g, n)$ is $(3, 1)$. (2) For any integers $g\geq 2$ and $n\geq 4$, there are infinitely many non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings of genus $g$.

On reducible Heegaard splittings

Let V∪SW be a reducible Heegaard splitting of genus g = g(S) ⩾ 2. For a maximal prime connected sum decomposition of V ∪SW, let q denote the number of the genus 1 Heegaard splittings of S2 × S1 in the decomposition, and p the number of all other prime factors in the decomposition. The main result of the present paper is to describe the relation of p, q and dim(CV ∩ CW).

Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Let M 1 and M 2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $${\phi : {\partial}M_1 \to {\partial}M_2}$$ . We analyze the relationship between the sets of low genus Heegaard splittings of M 1, M 2, and M, assuming the map $${\phi}$$ is “sufficiently complicated”. This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.

HIGH DISTANCE KNOTS IN CLOSED 3-MANIFOLDS

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus ≥2, then the coarse mapping class group of (S, V, W) is isomorphic to the mapping class group of (S, V, W).

Quasi-energy function for diffeomorphisms with wild separatrices

We consider the class G 4 of Morse—Smale diffeomorphisms on $$ \mathbb{S} $$ 3 with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse—Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class G 4,1 ⊂ G 4 of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere $$ \mathbb{S} $$ 3. For each diffeomorphism in G 4,1, we present a quasi-energy function with six critical points.

Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings

We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.

Heegaard splittings of twisted torus knots

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain “twisted torus knots” originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.

Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two

Let H 2 {\mathcal H_2} be the group of isotopy classes of orientation-preserving homeomorphisms of S 3 \mathbb S^3 that preserve a Heegaard splitting of genus two. In this paper, we construct a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of H 2 {\mathcal H_2} .

Examples of Lorentzian geodesible foliations of closed three-manifolds having Heegaard splittings of genus one

We construct codimension-one, Lorentzian geodesible foliations of closed three-manifolds having Heegaard splittings of genus one. We prove that all the inner leaves of a Reeb component of a codimension-one, totally geodesic foliation of a Lorentzian three-manifold are spacelike, and the boundary leaf of a Reeb component is lightlike.

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.