We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds.
The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem.
The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1.
In this note we prove that any closed graph manifold admitting a metric of non-positive sectional curvature (NPC-metric) has a finite cover, which is fibered over the circle. An explicit criterion to have a finite cover, which is fibered over the circle, is presented for the graph manifolds of certain class.
For a hyperbolic fibered 3-manifold M, we prove results that uniformly relate the structure of surface projections as one varies the fibrations of M. This extends our previous work from the fully-punctured to the general case.
We determine the rank of the fundamental group of those hyperbolic 3-manifolds fibering over the circle whose monodromy is a sufficiently high power of a pseudo-Anosov map. Moreover, we show that any two generating sets with minimal cardinality are Nielsen equivalent.
In "Tunnel one, fibered links", the second author showed that the tunnel of a tunnel number one, fibered link can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we analyze how the arc behaves under the monodromy action, and show that the tunnel arc is nearly clean, with the possible exception of twisting around the boundary of the fiber.
We determine the genus one fibered knots in lens spaces that have tunnel number one. We also show that every tunnel number one, once-punctured torus bundle is the result of Dehn filling a component of the Whitehead link in the 3-sphere.
The main result of this paper asserts that if a Seifert fibered 4-manifold has nonzero Seiberg-Witten invariant, the homotopy class of regular fibers has infinite order. This is a nontrivial obstruction to smooth circle actions; as applications, we show how to destroy smooth circle actions on a 4-manifold by knot surgery, without changing the integral homology, intersection form, and even the Seiberg-Witten invariant. Results concerning classification of Seifert fibered complex surfaces or symplectic 4-manifolds are included. We also show that every smooth circle action on the 4-torus is smoot…
By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whose 3-dimensional quotient Ω(F)/F has infinitely generated fundamental group, where Ω(F) is the discontinuity domain of F acting on the sphere at infinity. Moreover F is isomorphic to the fundamental group of a closed surface and contains infinitely many conjugacy classes of maxima…
We exhibit a closed aspherical 5-manifold of nonpositive curvature that fibers over a circle whose fundamental group is hyperbolic relative to abelian subgroups such that the fiber is a closed aspherical 4-manifold whose fundamental group is not hyperbolic relative to abelian subgroups.
We give a lower bound for the degree of a finite cover of a hyperbolic 3-manifold which fibers over the circle, in terms of volume, the diameter of the manifold and other new invariants.
Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topolog…
Suppose $α$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $α$ is Seifert in $H$(i.e., a 2-handle addition $H[α]$ is a Seifert-fibered space) and not the meridian of $H[R]$. Then for a slope $γ$ of $k$ represented by $α$, $γ$-Dehn surgery $k(γ)$ is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was i…
We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces diffeomorphic to the fiber. We also show, the existence of quasi-Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on t…
Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends toward a pair of projectively measured laminations that bind the surface, there is a convergent subsequence. This preprint also analyzes the quasi-isometric geometry of quasi-Fuchsian 3-manifolds. This eprint is based on a 1986 preprint, which was refereed and accepted for publication, but which I neglected to correct and return. The referee's corrections have…
It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered $3$-manifolds. We prove that there exist infinitely many closed orientable $3$-manifolds $M$ such that the minimal entropy over all hyperbolic, genus $g$ surface bundles over the circle as $2$-fold branched covers of the $3$-manifold $M$ is comparable to $1/g$.
The notion of a (stably) decomposable fiber bundle is introduced. In low dimensions, for torus fiber bundles over a circle the notion translates into a property of elements of the special linear group of integral matrices. We give a complete characterization of the stably decomposable torus fiber bundle of fiber-dimension less than 4 over the circle.
In this paper we study kleinian groups of Schottky type whose limit set is a wild knot in the sense of Artin and Fox. We show that, if the ``original knot'' fibers over the circle then the wild knot $Λ$ also fibers over the circle. As a consequence, the universal covering of $\mathbb{S}^{3}-Λ$ is $\mathbb{R}^{3}$. We prove that the complement of a dynamically-defined fibered wild knot can not be a complete hyperbolic 3-manifold.
We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays exponentially in the distance form the boundary of the convex core for points in the thick part. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves. We use this inflexibility to give a new proof of the convergence of pseudo-Anosov double-iteration on the quasi-Fuchsian space of a closed surface, and the r…
A Seifert manifold is a 3-dimensional manifold with a circle action. It is a circle bundle (with singularities) over a 2-dimensional orbifold. In this note, we discuss a generalized Seifert manifolds. By definition, they have bundle-like structures whose fibers are infra- homogeneous spaces; that is, the fibers are flat manifolds, almost flat manifolds, etc. We prove existence, uniqueness, rigidity theorems. Many interesting properties and applications are presented.
This paper initiates a systematic study of the relation of commensurability of surface automorphisms, or equivalently, fibered commensurability of 3-manifolds fibering over the circle. We show that every hyperbolic fibered commensurability class contains a unique minimal element, whereas the class of Seifert manifolds fibering over the circle consists of a single commensurability class with infinitely many minimal elements. The situation for non-geometric manifolds is more complicated, and we illustrate a range of phenomena that can occur in this context.
Murasugi sums can be defined as readily for Morse maps to the circle of (arbitrary) link complements in the 3-sphere as for fibrations over the circle of (fibered) link complements in the 3-sphere. As one application, I show that if a knot K has free genus m, then there is a Morse map from its complement to the circle (representing the relative homology class of a Seifert surface for K) with no more than 4m critical points.
We present a pair of smooth fiber bundles over the circle with a common $4$-dimensional fiber with the following properties: (1) their total spaces are diffeomorphic to each other; (2) they are isomorphic to each other as topological fiber bundles; (3) they are not isomorphic to each other as smooth fiber bundles. In particular, we exhibit an example with non-simply-connected fiber.
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume (whenever the manifold is hyperbolic), the dilatation of the monodromy, the injectivity radius, and the bottom eigenvalue of the Laplacian on these mapping tori. We also study mapping tori of punctured surface bundles, and various invariants of their Dehn fillings. We also study …
This is a "software upgrade" to a paper originally published in 1976, with cleaner statements and improved proofs. The main result is that, in a Haken 3-manifold, the space of all incompressible surfaces in a single isotopy class is contractible, except when the surface is the fiber of a surface bundle structure, in which case the space of all surfaces isotopic to the fiber has the homotopy type of a circle (the fibers). The main application from the 1976 paper is also rederived, the theorem (proved independently by Ivanov) that the diffeomorphism group of a Haken 3-manifold has contractible c…
We show that if the monodromy of a 3-manifold M that fibers over the circle has large translation distance in the curve complex, then the rank of the fundamental group of M is 2g+1, where g is the genus of the fiber.
We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact structure whose Euler class, when evaluated on the fiber, equals the Euler number of the fiber. This rigidity theorem is a consequence of properties of the action of pseudo-Anosov maps on the complex of curves of the fiber and a remarkable flexibility property of convex surfaces i…
Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic JSJ-subpair of M-S and show, in particular, that the guts of (M,S) is empty.
We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic.
In this paper it is proven that there is at most one way, up to isotopy, in which a connected, hyperbolic, orientable 3-manifold can fiber over the circle with monodromy in the Torelli group.
Any hyperbolic surface bundle over the circle gives rise to a continuous surjection from the circle to the sphere, by work of Cannon and Thurston. We prove that the order in which this surjection fills out the sphere is dictated by a natural triangulation of the surface bundle (introduced by Agol) when all singularities of the invariant foliations are at punctures of the fiber.
In this paper, we study the algebraic structure of mapping class group $Mod(X)$ of 3-manifolds $X$ that fiber as a circle bundle over a surface $S^1\rightarrow X\rightarrow S_g$. There is an exact sequence $1\rightarrow H^1(S_g)\rightarrow Mod(X)\rightarrow Mod(S_g)\rightarrow1$. We relate this to the Birman exact sequence and determine when this sequence splits.
The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3…
Using Hirasawa-Murasugi's classification of fibered Montesinos knots we classify the L-space Montesinos knots, providing further evidence towards a conjecture of Lidman-Moore that L-space knots have no essential Conway spheres. In the process, we classify the fibered Montesinos knots whose open books support the tight contact structure on $S^3$. We also construct L-space knots with arbitrarily large tunnel number and discuss the question of whether L-space knots admit essential tangle decompositions in the context of satellite operations and tunnel number.
By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for the Euler class (integer coefficients) of the normal bundle to a smooth circle action without fixed points on a manifold. In the special case of a 3-dimensional Seifert fibered space, this formula is particularly effective. \~
This article provides sufficient conditions for a closed hyperbolic 3-manifold $M$ with non zero first Betti number to fiber over the circle, and to find a fiber in $M$. Those conditions are formulated in terms of the behavior the circular characteristic in finite regular covers of $M$. We define the circular characteristic as an invariant associated to a non trivial cohomology class $α$ of $M$, using a Heegaard characteristic.
The main results of this paper describes a formula for the Seiberg-Witten invariant of a 4-manifold which admits a nontrivial free S^1-action. We use this theorem to produce a nonsymplectic 4-manifold with a free circle action whose orbit space fibers over S^1. We also describe a 3-manifold which is not the orbit space of any symplectic 4-manifold with a free circle action. A corollary of the main theorem is a formula for the 3-dimensional Seiberg-Witten invariants of the total space of a circle bundle over a surface.
This paper investigates certain foliations of three-manifolds that are hybrids of fibrations over the circle with foliated circle bundles over surfaces: a 3-manifold slithers around the circle when its universal cover fibers over the circle so that deck transformations are bundle automorphisms. Examples include hyperbolic 3-manifolds of every possible homological type. We show that all such foliations admit transverse pseudo-Anosov flows, and that in the universal cover of the hyperbolic cases, the leaves limit to sphere-filling Peano curves. The skew R-covered Anosov foliations of Sergio Fenl…
Maps between circle bundles: Fiber-preserving, Finiteness and Realization of mapping degree sets
University of Cyprus（塞浦路斯大学）；Rutgers University - New Brunswick（罗格斯大学-新不伦瑞克校区）；Academy of Mathematics and System Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Peking University（北京大学）
This paper is concerned with fixed-point free $S^1$-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) $S^1$-action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-p…
Given a torus bundle $Y$ over the circle and a cohomology class $[ω]\in H^2(Y;\mathbb{Z})$ which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of $Y$ with twisted coefficients in the universal Novikov ring.
Let $M$ be a closed 4-manifold with a free circle action. If the orbit manifold $N^3$ satisfies an appropriate fibering condition, then we show how to represent a cone in $H^2(M;\R)$ by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández-Gray-Morgan. In the case that $M$ is the product 4-manifold $S^1\times N$ our construction complements the results of \cite{FV08} (arXiv:0805:1234 [math.GT]) and allows us to completely determine the symplectic cone of such 4-manifolds. The content of this paper partly overlaps with the content of the unpublished prepr…
We propose a program to study groups acting faithfully on S^1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on S^1 is conjugate to a Fuchsian group if and only if it admits three very-full laminations with a variation of the transversality condition. Some partial results toward a similar characterization of hyperbolic 3-manifold groups which fiber…
We study the smallest positive eigenvalue $λ_1(M)$ of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold $M$ which fibers over the circle, with fiber a closed surface of genus $g\geq 2$. We show the existence of a constant $C>0$ only depending on $g$ so that $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2, C\log {\rm vol}(M)/{\rm vol}(M)^{2^{2g-2}/(2^{2g-2}-1)}]$ and that this estimate is essentially sharp. We show that if $M$ is typical or random, then we have $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2,C/{\rm vol}(M)^2]$. This rests on a result of independent interest about reccurence properties of axe…
We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke ei…
Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing a point from X. We prove that a subgroup G in H is convex cocompact in Mod(Y) if and only if G is finitely generated and purely pseudo-Anosov. We also prove a generalization of this theorem with H replaced by an arbitrary Gromov hyperbolic extension of the fundamental group of X, and an additional hypothesis of quasi-convexity of G in H. Along the way, we o…
