This paper is the second part of our work on 4-dimensional 2-handlebodies. In the first part (arXiv:math.GT/0407032) it is shown that up to certain set of local moves, connected simple coverings of B^4 branched over ribbon surfaces, bijectively represent connected orientable 4-dimensional 2-handlebodies up to 2-deformations (handle slides and creations/cancellations of handles of index <= 2). We factor this bijective correspondence through a map onto the closed morphisms in a universal braided category freely generated by a Hopf algebra object H. In this way we obtain a complete algebraic description of 4-dimensional 2-handlebodies. This result is then used to obtain an analogous description of the boundaries of such handlebodies, i.e. 3-dimensional manifolds, which resolves for closed manifolds the problem posed by Kerler in "Towards an algebraic characterization of 3-dimensional cobordisms", Contemporary Mathematics 318 (2003). (cf. Problem 8-16 (1) in T. Ohtsuki, "Problems on invariants of knots and 3-manifolds", Geom. Topol. Monogr. 4 (2002).
This paper gives a proof that the universal cover of a closed 3-manifold built from three $π_1$-injective handlebodies is homeomorphic to $\mathbb R^3$. This construction is an extension to handlebodies of the conditions for gluing of three solid tori to produce non-Haken Seifert fibered manifolds with infinite fundamental group. This class of manifolds has been shown to contain non-Haken non-Seifert fibered manifolds.
This paper looks at a class of closed orientable 3-manifolds constructed from a gluing of three handlebodies, such that the inclusion of each handlebody is $π_1$-injective. This construction is the generalisation to handlebodies of the condition for gluing three solid tori to produce non-Haken Seifert fibered 3-manifolds with infinite fundamental group. It is shown that there is an efficient algorithm to decide if a gluing of handlebodies meets the disk-condition. Also an outline for the construction of the characteristic variety (JSJ decomposition) in such manifolds is given. Some non-Haken a…
This paper gives a proof that the fundamental group of a class of closed orientable 3-manifolds constructed from three injective handlebodies has a solvable word problem. This is done by giving an algorithm to decide if a closed curve in the manifold is null-homotopic. Non-Haken and non-Seifert fibered examples are constructed by performing Dehn surgery on a class of two-bridge knots.
The main theorem of this paper is a generalisation of well known results about Dehn surgery to the case of attaching handlebodies to a simple 3-manifold. The existence of a finite set of `exceptional' curves on the boundary of the 3-manifold is established. Provided none of these curves is attached to the boundary of a disc in a handlebody, the resulting manifold is shown to be word hyperbolic and `hyperbolike'. We then give constructions of gluing maps satisfying this condition. These take the form of an arbitrary gluing map composed with powers of a suitable homeomorphism of the boundary of …
Work of numerous authors has shown that any smooth, orientable, closed 4-manifold may be described as a loop of Morse functions on a surface, a loop in the cut complex, a loop in the pants complex, or as a multisection. In this paper, we prove a corresponding uniqueness theorem for each of these descriptions, so that, for example, any two loops of Morse functions on a surface yielding diffeomorphic 4-manifolds are related by a given set of moves.
We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.
Automorphisms of handlebodies arise naturally in the a classification of automorphisms of three-manifolds. Among automorphisms of handlebodies, there are certain automorphisms called irreducible (or generic), which are analogues of pseudo-Anosov automorphisms of surfaces. We show that irreducible automorphisms of handlebodies exist and develop methods for constructing a range of examples.
We establish a bijective correspondence between the set T(n) of 3-dimensional triangulations with n tetrahedra and a certain class H(n) of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n+1. We show that the manifolds in H(n) are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops. Mirroring the elements of H(n) in their geodesic boundary we obtain a class D(n) of cusped hyperbolic manifolds, previously considered by D. Thurston and the first named author. We show tha…
In this paper, we consider decompositions of 3-manifolds with three handlebodies. We classify such decompositions of the 3-sphere and lens spaces with small genera. These decompositions admit operations called stabilizations. We also determine whether these decompositions are stabilized.
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.
Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations as a key tool. In particular, we prove that every closed piecewise linear $n$-manifold has a multisection, i.e. can be divided into $k+1$ $n$-dimensional $1$-handlebodies, where $n=2k+1$ or $n=2k$, such that intersections of the handlebodies have spines of small dimensions. Several applications, constructions and generalisations of our approach are given.
Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over $S^2$.
In 2015, Chen and Yang proposed a volume conjecture that stated that certain Turaev-Viro invariants of an hyperbolic 3-manifold should grow exponentially with a rate equal to the hyperbolic volume. Since then, this conjecture has been proven or numerically tested for several hyperbolic 3-manifolds, either closed or with boundary, the boundary being either a family of tori or a family of higher genus surfaces. The current paper now provides new numerical checks of this volume conjecture for 3-manifolds with one toroidal boundary component and one geodesic boundary component. More precisely, we …
We consider braids on $m+n$ strands, such that the first $m$ strands are trivially fixed. We denote the set of all such braids by $B_{m,n}$. Via concatenation $B_{m,n}$ acquires a group structure. The objective of this paper is to find a presentation for $B_{m,n}$ using the structure of its corresponding pure braid subgroup, $P_{m,n}$, and the fact that it is a subgroup of the classical Artin group $B_{m+n}$. Then we give an irredundant presentation for $B_{m,n}$. The paper concludes by showing that these braid groups or appropriate cosets of them are related to knots in handlebodies, in knot …
We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, which provides a complete solution to the long-standing Fox-Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Ap…
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…
In this paper, we consider decompositions of closed orientable 3-manifolds with more than 3 handlebodies, where the union of intersections of handlebodies is a multibranched surface. We define stabilization operations for such decompositions and show the stable equivalence.
Let F a closed connected orientable surface bounding a genus g handlebody H. In this paper we find a finite set of generators for the subgroup E(2,g) of the pure mapping class group of the twice punctured torus PMCG(2,g), consisting of the isotopy classes of homeomorphisms of F which admit an extension to H keeping a properly embedded trivial arc fixed. This subgroup turns out to be important for the study of knots in closed 3-manifolds via (g,1)-decomposition. In fact, the knots represented by the isotopy classes belonging to the same left cosets of E(2,g) in PMCG(2,g) are equivalent.
We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev's shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is. We characterise here all the closed oriented 4-manifolds that have complexity at most one. They are generated by a certain set of 20 blocks, that are some basic 4-manifolds with boundary consisting of copies of $S^2 \times S^1$, plus connected sums with some copies of $\mathbb{CP}^2$ with either orientation. All the manifolds generated by these blocks are doubles. Many …
For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a ``hyperelliptic'' three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the {\sl Weierstrass lines} of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierst…
Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $Σ$, guiding the gluing of the handlebodies. Any morphism $φ$ from $π_1(X)$ to a finitely generated free abelian group induces a morphism on $π_1(Σ)$. We express the twisted homology and Reidemeister torsion of $(X;φ)$ in terms of the first homology of $(Σ;φ)$ and the three subspaces generated by the collections of curves. We also express the intersec…
Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the skein module of this 3-mani…
Given a symmetry $τ$ of a closed Riemann surface $S$, there exists an extended Kleinian group $K$, whose orientation-preserving half is a Schottky group $Γ$ uniformizing $S$, such that $K/Γ$ induces $\langle τ\rangle$; the group $K$ is called an extended Schottky group. A geometrical structural description, in terms of the Klein-Maskit combination theorems, of both Schottky and extended Schottky groups is well known. A dihedral extended Schottky group is a group generated by the elements of two different extended Schottky groups, both with the same orientation-preserving half. Such configurati…
The main theorem of this paper generalizes recent results in Dehn surgery to the case of handlebody attachment. We consider attaching handlebodies and solid tori to the boundary of an irreducible, boundary-irreducible, atoroidal and acylindrical 3-manifold. We show that for a large class of homeomorphisms attaching these handlebodies, the fundamental group of the resulting manifold contains the fundamental group of a closed surface of genus at least two.
We define a trisection of a closed, orientable three dimensional manifold into three handlebodies, and a notion of stabilization for these trisections. Several examples of trisections are described in detail. We define the trisection genus $t(M)$ of a 3-manifold, and relate it to the Heegaard genus $g(M)$, showing that $t(M) \le g(M) \le 2t(M)$. We show moreover that the bound $g(M) \le 2t(M)$ is tight. We define stabilizations of trisections and show that all trisections of a 3-manifold are stably equivalent, providing an analogue of the Reidemeister-Singer theorem for trisections. We conclud…
We show that a smooth embedding of a closed 3-manifold in S^3 x R can be isotoped so that every generic level divides S^3 x t into two handlebodies (i.e., is Heegaard) provided the original embedding has a unique local maximum with respect to the R coordinate. This allows uniqueness of embeddings to be studied via the mapping class group of surfaces and the Schoenflies conjecture is considered in this light. We also give a necessary and sufficient condition that a 3-manifold connected summed with arbitrarily many copies of S^1 x S^2 embeds in R^4.
We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, in terms of local moves. This result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of our equivalence theorem to possibly non-simple coverings of S^3 br…
Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a $4$-dimensional manifold as a union of three $4$-dimensional handlebodies. The complexity of the $4$-manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or $4$-dimensional simplice…
Using handlebodies we verify a conjecture of P Adjamagbo that if the frontier of a relatively compact subset $V_0$ of a manifold is a submanifold then there is an increasing family $\{V_r\}$ of relatively compact open sets indexed by the positive reals so that the frontier of each is a submanifold, their union is the whole manifold and for each $r\ge 0$ the subfamily indexed by $(r,\infty)$ is a neighbourhood basis of the closure of the $r^{\rm th}$ set.
We show that if $M$ is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of $M$ is finite.
We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$, i.e. the classifying space $BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near a $2n$-dimensional disk embedded in the boundary, $\partial(D^{n+1}\times S^n)^{\natural g}$. We construct a map $colim_{g\to\infty}BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \longrightarrow Q_{0}BO(2n+1)\langle n \rangle_{+}$ and prove that it induces an isomorphism on integral homology in the case that $2n+1 \geq 9$. Above, $BO(2n+1)\langle n \rangle…
We show that any smooth, closed, oriented, connected 4--manifold can be trisected into three copies of $\natural^k (S^1 \times B^3)$, intersecting pairwise in 3--dimensional handlebodies, with triple intersection a closed 2--dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3--manifolds. A trisection of a 4--manifold $X$ arises from a Morse 2--function $G:X \to B^2$ and the obvious trisection of $B^2$, in much the same way that a Heegaard splitting of a 3--m…
In this paper we prove a homological stability theorem for the diffeomorphism groups of high dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product $D^{p+1}\times S^{q}$ for $|q - p| < \min\{p, q\} - 2$. In a recent joint paper with Boris Botvinnik (see arXiv:1509.03359 ), we identify the homology of $colim_{g\to \infty}BDiff((D^{n+1}\times S^{n})^{\natural g}, \; D^{2n})$ with that of the infinite loopspace $Q_{0}BO(2n+1)\langle n\rangle_{+}$, in the case that $n \geq 4$. Combining this "stable homology" calculation with this paper's homologic…
Multisections and bridge positions in arbitrary dimensions
Institut Fourier, Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学傅里叶研究所）；Department of Mathematics, University of California, Berkeley（加州大学伯克利分校数学系）；Department of Mathematical Sciences, Ritsumeikan University（立命馆大学数理科学系）
Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold Y can be embedded in M so that the exterior of the image of Y is a union of handlebodies; and (ii) a compact connected 3-manifold Y can be embedded in M so that every knot in M can be isotoped to lie within the image of Y . Our result can be regarded as a common generalization of the reimbedding theorem by Fox [Fox48] and the charact…
This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction th…
The main theorem of this article provides sufficient conditions for a degree $d$ finite cover $M'$ of a hyperbolic 3-manifold $M$ to be a surface-bundle. Let $F$ be an embedded, closed and orientable surface of genus $g$, close to a minimal surface in the cover $M'$, splitting $M'$ into a disjoint union of $q$ handlebodies and compression bodies. We show that there exists a fiber in the complement of $F$ provided that $d$, $q$ and $g$ satisfy some inequality involving an explicit constant $k$ depending only on the volume and the injectivity radius of $M$. In particular, this theorem applies to…
We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant). Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspheri…
The aim of this article is to introduce and study certain topological invariants for closed, oriented three-manifolds Y. These groups are relatively Z-graded Abelian groups associated to SpinC structures over Y. Given a genus g Heegaard splitting of Y, these theories are variants of the Lagrangian Floer homology for the g-fold symmetric product of the surface relative to certain totally real subspaces associated to the handlebodies.
Let k be an integral domain containing the invertible elements α, s and \frac{1}{s-s^{-1}}. If M is an oriented 3-manifold, let K(M) denote the Kauffman skein module of M over k. Based on the work on Birman-Murakami-Wenzl algebra by Beliakova and Blanchet, we give an ``idempotent-like'' basis for the Kauffman skein module of handlebodies. Gilmer and Zhong have studied the Homflypt skein modules of a connected sum of two 3-manifolds, here we study the case for the Kauffman skein module and show that K(M_1 # M_2) is isomorphic to K(M_1) tensor K(M_2) over a certain localized ring, where M_1 # M_…
We demonstrate how to combinatorially calculate the EH-class of a compatible contact structure in the sutured Floer homology group of a balanced sutured three manifold which is associated to an abstract partial open book decomposition. As an application we show that every contact three manifold (closed or with convex boundary) can be obtained by gluing tight contact handlebodies whose EH-classes are nontrivial.
Donaldson showed that every closed symplectic 4-manifold can be given the structure of a topological Lefschetz pencil. Gay and Kirby showed that every closed 4-manifold has a trisection. In this paper we relate these two structure theorems, showing how to construct a trisection directly from a topological Lefschetz pencil. This trisection is such that each of the three sectors is a regular neighborhood of a regular fiber of the pencil. This is a 4-dimensional analog of the following trivial 3-dimensional result: For every open book decomposition of a 3-manifold M, there is a decomposition of M…
(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 t…
Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In…
