3-dimensional Handlebody Research Articles

A standard form of incompressible surfaces in 3-dimensional handlebodies

Applying the Morse theory, we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies. We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form. Our algorithm is practical. Several examples are given to test the algorithm.

Group trisections and smooth 4–manifolds

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4-manifold. Together with Gay and Kirby's existence and uniqueness theorem for 4-manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4-manifolds modulo diffeomorphism. As a consequence, smooth 4-manifold topology is, in principle, entirely group theoretic. For example, the smooth 4-dimensional Poincar\'e conjecture can be reformulated as a purely group theoretic statement.

A twisted first homology group of the handlebody mapping class group

Let $H_g$ be a 3-dimensional handlebody of genus $g$. We determine the twisted first homology group of the mapping class group of $H_g$ with coefficients in the first integral homology group of the boundary surface $\partial H_g$ for $g\ge2$.

Self-affine manifolds

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile R3. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to higher dimensions are established. The methods developed build a bridge linking geometric topology with iterated function systems and their attractors.A method to model self-affine tiles by simple iterative systems is developed in order to study their topology. The model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self-affine tile. It has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold. The induced map between the attractor of the model and the self-affine tile is a quotient map and can be checked in certain cases to be monotone or cell-like. Deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self-affine tile is a topological or generalized manifold in all dimensions. These new tools are used to check that several self-affine tiles in the literature are 3-balls. An example of a wild 3-dimensional self-affine tile is given whose boundary is a topological 2-sphere but which is not itself a 3-ball. The paper describes how any 3-dimensional handlebody can be given the structure of a self-affine 3-manifold. It is conjectured that every self-affine tile which is a manifold is a handlebody.

On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus $g>1$ is bounded by the linear polynomial $12(g-1)$, and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalen

Degree of roots of disk twists on 3-dimensional handlebodies

Margalit and Schleimer (Geom Topol 13(3):1495–1497, 2009) discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. McCullough and Rajeevsarathy (Geom Dedicata 151(1):397–409, 2011) and Monden (Rocky Mt J Math, to appear) obtained the evaluation of the degrees of roots of Dehn twists. In this paper, we discuss existence and degrees of homeomorphisms whose power is equal to disk twist about a nonseparating disk in the mapping class group of the 3-dimensional handlebody.

The Green’s functions of the boundaries at infinity of the hyperbolic 3-manifolds

The work is motivated by a result of Manin in [1], which relates the Arakelov Green’s function on a compact Riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with Schottky uniformization, having the Riemann surface as a conformal boundary at infinity. A natural question is to what extent the result of Manin can be generalized to cases where, instead of dealing with a single Riemann surface, one has several Riemann surfaces whose union is the boundary of a hyperbolic 3-manifold, uniformized no longer by a Schottky group, but by a Fuchsian, quasi-Fuchsian, or more general Kleinian group. We have considered this question in this work and obtained several partial results that contribute towards constructing an analog of Manin’s result in this more general context.

The framed little 2-discs operad and diffeomorphisms of handlebodies

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (that is, the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ‘open string’ modular operad made from moduli spaces of Riemann surfaces with marked intervals on the boundary. Our technique also recovers a theorem of Braun that the derived modular envelope of the cyclic operad that describes associative algebras with involution is homotopy equivalent to the modular operad made from moduli spaces of unoriented Klein surfaces with open string gluing.

Abelianization and Nielsen realization problem of the mapping class group of a handlebody

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms to the mapping class group has no section when g is at least 5. In order to prove the above result, we show the vanishing of the first homology group with the real coefficient of the mapping class group of the handlebody with genus g at least 3 and a distinguished disk on its boundary.

Involutions of 3-dimensional handlebodies

We study the orientation preserving involutions of the orientable 3-dimensional handlebody $H_g$, for any genus $g$. A complete classification of such involutions is given in terms of their fixed points.

Realization of the mapping class group of handlebody by diffeomorphisms

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching 1-handles, it is shown that the natural surjection from the group of orientation-preserving diffeomorphisms of it to the mapping class group of it has no section when is at least 6.

Heegaard Splittings and Pseudo-Anosov Maps

Given two 3-dimensional handlebodies whose boundaries are identified with a surface S of genus g > 1 and with different orientations, we consider the sequence of manifolds Mn obtained by gluing the handlebodies via the iteration fn of a “generic” pseudo-Anosov homeomorphism f of S. Using the deformation theory of hyperbolic structures on open hyperbolic 3-manifolds and for n sufficiently large, we construct a negatively curved metric on Mn where the sectional curvatures are pinched in a given small interval centered at –1. The construction is concrete enough to allow us describe the geometric limits of these manifolds as n tends to infinity and the metrics get closer to being hyperbolic. Such a description allows us to prove various topological and group theoretical properties of Mn, for n sufficiently large, which would not be available knowing the mere existence of a negatively curved or even hyperbolic metric on Mn.

Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of V is a handlebody of a Heegaard splitting of X. Provided that the genus of V is at least 2, there are similar choices with X closed and hyperbolic.

On non-singular stable maps of 3-manifolds with boundary into the plane

Let M be a compact connected orientable 3-manifold with non-empty boundary and f : M ! R a stable map. In this paper we study the existence of an immersion or embedding lift of f to R nV 3† with respect to the standard projection R ! R. We also characterize the orientable 3-dimensional handlebody in terms of stable maps which have only a restricted class of singularities. Moreover, by using the concept of an embedding lift of a certain map of a 2-dimensional polyhedron into R, we give a characterization of S.

Mapping class group of a handlebody

Let B be a 3-dimensional handlebody of genus g. Let M be the group of the isotopy classes of orientation preserving homeomorphisms of B. We construct a 2-dimensional simplicial complex X, connected and simply-connected, on which M acts by simplicial transformations and has only a finite number of orbits. From this action we derive an explicit finite presentation ofM. We consider a 3-dimensional handlebody B = Bg of genus g > 0. We may think of B as a solid 3-ball with g solid handles attached to it (see Figure 1). Our goal is to determine an explicit presentation of the mapping class group of B, the group Mg of the isotopy classes of orientation preserving homeomorphisms of B. Every homeomorphism h of B induces a homeomorphism of the boundary S = ∂B of B and we get an embedding of Mg into the mapping class group MCG(S) of the surface S.

Extending finite group actions from surfaces to handlebodies

We show that every action of a finite dihedral group on a closed orientable surface F \mathcal F extends to a 3-dimensional handlebody V \mathcal V , with ∂ V = F \partial \mathcal V=\mathcal F . In the case of a finite abelian group G G , we give necessary and sufficient conditions for a G G -action on a surface to extend to a compact 3 3 -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order 84 ( g − 1 ) 84(g-1) of P S L ( 2 , 27 ) PSL(2,27) on a surface F \mathcal F of genus g = 118 g=118 does not extend to any compact 3-manifold M M with ∂ M = F \partial M=\mathcal F , thus resolving the only case of Hurwitz actions of type P S L ( 2 , q ) PSL(2,q) of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).

A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth

Let M be a closed, orientable 3-manifold. We say that (V, V'\F) is a Heegaard splitting of M if V, V are 3-dimensional handlebodies in M such that M = F U F VnV'=dV=dV'=F. Then F is called a Heegaard surface of M, and the genus of F is called the genus of the Heegaard splitting. We say that two Heegaard splittings (V,V; F) and (W, W' G) of M are homeomorphic if there is a self-homeomorphism / of M such that f(F)—G. Then we denote the number of the homeomorphism classes of the Heegaard splittings of genus g of M by hM(g) (possibly hM(g)= oo). We note that K. Johannson showed that if M is a Haken manifold, then hM(g) is finite for every g [8], [9]. In [12], F. Waldhausen asked whether hM{2g)= provided M admits a Heegaard splitting of genus g. Casson-Gordon showed that the answer to this question is No. In fact they showed that there exist infinitely many 3-manifolds M such that hM(2ή)>2 for all n>3 [5]. In this paper, we improve this result as follows.

Finite group actions on handlebodies and equivariant Heegaard genus for 3-manifolds

Every finite group of symmetries (homeomorphisms) of a compact bounded surface of algebraic genus g acts, by taking the product with the interval [0,1], also on the 3-dimensional handlebody V g of genus g. In both cases, the maximal possible order of such a group is 12( g−1), and we call such a group a maximal bounded surface respectively handlebody group. Here we construct the first examples of maximal handlebody groups which are not maximal bounded surface groups. Our examples lead in a natural way to the notion of an equivariant Heegaard genus for finite group actions on 3-manifolds; we compute this genus for some classes of interesting examples. The notion of an equivariant Heegaard genus gives a certain hierarchy for finite group actions on a 3-manifold and the notion of a maximally symmetric 3-manifold; we show that the irreducible maximally symmetric 3-manifolds belong to the hyperbolic geometry, or are Seifert fibered of a very special type.

The Symmetries of Genus One Handlebodies

The symmetries of manifolds are a focal point of study in low-dimensional topology and yet, outside of some totally asymmetrical 3- and 4-manifolds, there are very few cases in which a complete classification has been attained. In this work we provide such a classification for symmetries of the orientable and nonorientable 3-dimensional handlebodies of genus one. Our classification includes a description, up to isomorphism, of all of the finite groups which can arise as symmetries on these manifolds, as well as an enumeration of the different ways in which they can arise. To be specific, we will classify the equivalence, weak equivalence and strong equivalence classes of (effective) finite group actions on the genus one handlebodies.

Incompressible Surfaces in the Boundary of a Handlebody–an Algorithm

Our first result is a decomposition theorem for free groups relative to a set of elements. This enables us to formulate several algebraic conditions, some necessary and some sufficient, for various surfaces in the boundary of a 3-dimensional handlebody to be incompressible. Moreover, we show that there exists an algorithm to determine whether or not these algebraic conditions are met.Many of our algebraic ideas are similar to those of Shenitzer [3]. Conversations with Professor Roger Lyndon were helpful in the initial development of these results, and he reviewed an earlier version of this paper, suggesting Theorem 1 (iii) and its proof. Our notation and techniques are standard (cf. [1], [2]). A set X of elements in a finitely generated free group F is a basis if it is a minimal generating set, and X±l denotes the set of all elements in X, together wTith their inverses.