Handlebody Of Genus Research Articles

On $ H' $-splittings of a handlebody

<p>Let $ M $ be a compact connected orientable 3-manifold and $ F $ be a compact connected orientable surface properly embedded in $ M $. If $ F $ cuts $ M $ into two handlebodies $ X $ and $ Y $ (i.e., $ M = X\cup_FY $), then we say that $ F $ is an $ H' $-splitting surface for $ M $ and call $ X\cup_FY $ an $ H' $-splitting for $ M $. When the $ H' $-splitting surface $ F $ is incompressible in a handlebody $ H $, a characteristic of an $ H' $-splitting $ H_1\cup_F H_2 $ to denote $ H $ is already known. In the present paper, we generalize the above result as follows: Let $ H $ be a handlebody of genus $ g\geq 1 $, $ X\cup_F Y $ an $ H' $-splitting for $ H $. Then, either $ X\cup_F Y $ is stabilized, or there exists a reducing system $ \mathcal{J}_1\cup\mathcal{K}_1 $ of $ F $, such that $ \mathcal{J}_1 $ is quasi-primitive in $ Y $ and $ \mathcal{K}_1 $ is quasi-primitive in $ X $. Combining the result with the known result, we obtain a characteristic of an $ H' $-splitting $ H_1\cup_F H_2 $ to denote a handlebody.</p>

Tied links in various topological settings

Tied links in [Formula: see text] were introduced by Aicardi and Juyumaya as standard links in [Formula: see text] equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces [Formula: see text], in handlebodies of genus [Formula: see text], and in the complement of the [Formula: see text]-component unlink. We introduce the tied braid monoids [Formula: see text] by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an [Formula: see text]-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

Equivalence of ℤ4-actions on Handlebodies of Genus g

In this paper we consider all orientation-preserving $\mathbb{Z}_{4}$-actions on $3$-dimensional handlebodies $V_g$ of genus $g>0$. We study the graph of groups $(\Gamma($v$),\mathbf{G(v)})$, which determines a handlebody orbifold $V(\Gamma($v$),{\mathbf{G(v)}})\simeq{V_g}/\mathbb{Z}_{4}$. This algebraic characterization is used to enumerate the total number of $\mathbb{Z}_{4}$ group actions on such handlebodies, up to equivalence.

On the connectivity of the branch locus of the Schottky space

Let M be a handlebody of genus g greater than= 2. The space T(M), that parametrizes marked Kleinian structures on M up to isomorphisms, can be identified with the space MSg, of marked Schottky groups of rank g, so it carries a structure of complex manifold of finite dimension 3(g - 1). The space M(M) parametrizing Kleinian structures on M up to isomorphisms, can be identified with S-g, the Schottky space of rank g, and it carries the structure of a complex orbifold. In these identifications, the projection map pi: T(M) -greater than M(M) corresponds to the map from MSg, onto S-g that forgets the marking. In this paper we observe that the singular locus B(M) of M(M), that is, the branch locus of pi, has (i) exactly two connected components for g = 2, (ii) at most two connected components for g greater than= 4 even, and (iii) M(M) is connected for g greater than= 3 odd.

Finite-sided deformation spaces of complete affine 3-manifolds

A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic to a two-holed cross-surface, that is, the complement of two disjoint discs in the real projective plane. We show that every such manifold is homeomorphic to a solid handlebody of genus two, and admits a fundamental polyhedron bounded by crooked planes. Furthermore, the deformation space is a bundle of convex quadrilateral cones over the space of marked hyperbolic structures. The sides of each quadrilateral cone are defined by invariants of the two boundary components and the two orientation-reversing simple curves. The two-holed cross-surface, together with the three-holed sphere, are the only topologies for which the deformation space of complete affine structures is finite-sided.

Fixed points of imaginary reflections on hyperbolic handlebodies

AbstractA Klein–Schottky group is an extended Kleinian group, containing no reflections and whose orientation-preserving half is a Schottky group. A dihedral-Klein–Schottky group is an extended Kleinian group generated by two different Klein–Schottky groups, both with the same orientation-preserving half. We provide a structural description of the dihedral-Klein–Schottky groups.Let M be a handlebody of genus g, with a Schottky structure. An imaginary reflection τ of M is an orientation-reversing homeomorphism of M, of order two, whose restriction to its interior is an hyperbolic isometry having at most isolated fixed points. It is known that the number of fixed points of τ is at most g + 1; τ is called a maximal imaginary reflection if it has g + 1 fixed points. As a consequence of the structural description of the dihedral-Klein–Schottky groups, we are able to provide upper bounds for the cardinality of the set of fixed points of two or three different imaginary reflections acting on a handlebody with a Schottky structure. In particular, we show that maximal imaginary reflections are unique.

Handle additions producing essential surfaces

We construct a small, hyperbolic 3-manifold M with the property that, for any integer g = 2, there are infinitely many separating slopes r in ?M such that the 3-manifold M(r) obtained by attaching a 2-handle to M along r contains an essential separating closed surface of genus g. The resulting manifolds M(r) are still hyperbolic. This contrasts sharply with known finiteness results on Dehn filling and with the known finiteness result on handle addition for the cases g = 0,1. Our 3-manifold M is the complement of a hyperbolic, small knot in a handlebody of genus 3.

Singular-Hyperbolic Attractors with Handlebody Basins

An attractor of a vector field X with generating flow X t is a transitive set equal to $${\bigcap\limits_{t > 0} {X_{t} {\left( U \right)}} }$$ for some neighborhood U called basin of attraction. An attractor is singular-hyperbolic if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding direction. A singular-hyperbolic repeller is a singular-hyperbolic attractor for the time reversed flow. A handlebody of genus $$n \in \mathbb{N}$$ is a compact 3-manifold V containing a disjoint collection of n properly embedded 2-cells such that the result of cutting V along these disks is a 3-cell. We show that every orientable handlebody of genus n???2 can be realized as the basin of attraction of a singular-hyperbolic attractor. Hence every closed orientable 3-manifold supports a vector field whose nonwandering set consists of a singular-hyperbolic attractor and a singular-hyperbolic repeller. In particular, there are open sets of C r vector fields without hyperbolic attractors or hyperbolic repellers on every closed 3-manifold, r???1.

ON MAXIMAL COLLECTIONS OF ESSENTIAL ANNULI IN A HANDLEBODY

It is known that, for a maximal collection [Formula: see text] of pairwise disjoint non-parallel essential annuli in a handlebody of genus 2, [Formula: see text]. We show that for a maximal collection [Formula: see text] of pairwise disjoint non-parallel essential annuli in a handlebody of genus n (≥ 3), [Formula: see text], and the bound is best possible.

Simple, small knots in handlebodies

We construct a class of simple, small knots in a handlebody of genus g, for any g>1.

KNOT THEORY IN HANDLEBODIES

We consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in S3, which are elements of the braid groups Bg,n. We prove a geometric version of the Markov theorem for braid equivalence in the handlebody, which is based on the L-moves. Using this we then prove two algebraic versions of the Markov theorem. The first one uses the L-moves. The second one uses the Markov moves and conjugation in the groups Bg,n. We show that not all conjugations correspond to isotopies.

Dehn surgery equivalence relations on 3-manifolds

Suppose M is an oriented 3-manifold. A Dehn surgery on M (defined below) is a process by which M is altered by deleting a tubular neighbourhood of an embedded circle and replacing it again via some diffeomorphism of the boundary torus. It was shown by Lickorish [Li] and Wallace [Wa] that any closed oriented connected 3-manifold can be obtained from any other such manifold by a finite sequence of Dehn surgeries. Thus under this equivalence relation all closed oriented 3-manifolds are equivalent. We shall investigate this same question for more restricted classes of surgeries. In particular we shall insist that our Dehn surgeries preserve the integral (or rational) homology groups. Specifically, if M0 and M1 have isomorphic integral (respectively rational) homology groups, is there a sequence of Dehn surgeries, each of which preserves integral (respectively rational) homology, that transforms M0 to M1? What is the situation if we further restrict the Dehn surgeries to preserve more of the fundamental group? Is there a difference if we require ‘integral’ surgeries? We also show that these Dehn surgery relations are strongly connected to the following questions concerning another point of view towards understanding 3-manifolds. Is there a Heegard splitting of M0, M0 = H1 ∪fH2 (Hi are handlebodies of genus g and f is a homeomorphism of their common boundary surface), and a homeomorphism g of ∂H1 such that M1 has a Heegard splitting using g ∘ f as the identification? Since there are many natural subgroups of the mapping class group, such as the Torelli subgroup and the ‘Johnson subgroup’, one can ask the same question where g is restricted to lie in one of these subgroups. This is related to work of Morita on Casson's invariant for homology 3-spheres [Mo1]. Even under these restrictions it has been known for some time that any homology 3-sphere is related to S3. This fact has been used to define, calculate and understand invariants of homology 3-spheres (such as Casson's invariant) by choosing such a ‘path to S3’ in the ‘space’ of 3-manifolds.

Incompressible surfaces in handlebodies and closed 3-manifolds of Heegaard genus 2

In this paper, we shall prove that for any integer n > 0, 1) a handlebody of genus 2 contains a separating incompressible surface of genus n, 2) there exists a closed 3-manifold of Heegaard genus 2 which contains a separating incompressible surface of genus n.

Finite maximal orientation reversing group actions on handlebodies and 3-manifolds

We study arbitrary (that is not necessarily orientation preserving) finite group actions on 3-dimensional orientable or nonorientable handlebodies of genus g. For g>1, the maximal possible order is 24(g−1); we characterize the corresponding groups of this order and also the occuring quotient orbifolds. Then we use this to study finite group actions of large order (with respect to the equivariant Heegaard genus g) on closed 3-manifolds, again concentrating on the maximal case of order 24(g−1). Our results extend corresponding results in the orientation preserving setting. Whereas for arbitrary finite group actions on handlebodies much more types of quotient orbifolds occur than in the orientation preserving case, it turns out that for closed 3-manifolds the situation is quite rigid, in contrast to the orientation preserving case where one has many possibilities to construct manifolds with large group actions.

Generating Functions for Actions on Handlebodies with Genus Zero Quotient

For a finite group G and a nonnegative integer g, let Q g denote the number of q-equivalence classes of orientation-preserving G-actions on the handlebody of genus g which have genus zero quotient. Let q( z)=∑ g⩾0 Q g z g be the associated generating function. When G has at most one involution, we show that q( z) is a rational function whose poles are roots of unity. We prove a partial converse showing that when G has more than one involution, q( z) is either irrational or has a pole in the open disk {| z|<1}. In the case where G has at most one involution, we obtain an asymptotic approximation for Q g by analyzing a finite poset which embodies information about generating multisets of G. A finer approximation is found when G is cyclic.

On a hyperbolic 3-manifold with some special properties

We present a closed hyperbolic 3-manifold M with some surprising properties. The universal covering group of M is a normal torsion-free subgroup of minimal index in one of the nine Coxeter groups G, generated by the reflections in the faces of one of the nine Lannér-tetrahedra (bounded tetrahedra in hyperbolic 3-space all of whose dihedral angles are of the form π/n with n ∈ ℕ see [1] or [3]). The corresponding Coxeter group G splits as a semidirect product G = π1M⋉A, where A is a finite subgroup of G, and G is the only one of the nine Coxeter groups associated to the Lannér-tetrahedra which admits such a splitting (this follows using results in [4]). We derive a presentation of π1M and show that the first homology group H1(M) of M is isomorphic to ℚ11. This is in sharp contrast to other torsion-free (non-normal) subgroups of finite index in Coxeter groups constructed in [1] which all have finite first homology (though it is known that they are all virtually ℚ-representable (see [5], p. 434). It follows from our computations that the Heegaard genus of M is 11, and that there exists a Heegaard splitting of M of genus 11 invariant under the action of the group I+(M) ≌ S5 ⊕ ℚ2 of orientation-preserving isometries of M (we compute this group in [4]), so that the Heegaard genus of M is equal to the equivariant Heegaard genus of the action of I+(M) on M. Moreover M is maximally symmetric in the sense of [4, 6]: the order 120 of the subgroup of index 2 in I+(M) which preserves both handle-bodies of the Heegaard splitting is the maximal possible order of a group of orientation-preserving diffeomorphisms of a handle-body of genus 11. (This maximal order is 12(g—1) for a handle-body of genus g; see [7].) By taking the coverings Mq of M corresponding to the surjections π1M→H1(M) ≌ ℚ11→(ℚq)11 for q ∈ ℕ, we obtain explicitly an infinite series of maximally symmetric hyperbolic 3-manifolds.

Minimal atlases on 3-manifolds

AbstractIn this paper we investigate the minimal number of charts (spaces homeomorphic to open subsets of ℝ3) needed to cover a closed 3-manifold M3. We prove that this number is two if the Bockstein βω1(M3)∈H2(M3;ℤ) of the first Stiefel-Whitney class of M3 is zero and three if it is not. We state several properties equivalent to βω1(M3) = 0, for example the condition of being covered by two orientable subspaces, and the condition that all torsion elements of H1(M3;ℤ) be represented by orientation-preserving loops. We also show that the non-orientable 3-manifolds which can be covered with two charts are precisely those manifolds which can be described as sewn-up r-link exteriors. These are manifolds obtained by removing from S3 the interiors of two disjoint handlebodies of genus r and then identifying the two boundary components by a homeomorphism.