Real places and torus bundles

Virtual homology of surgered torus bundles

Let $M$ be a once-punctured torus bundle over $S^1$ with monodromy $h$. We show that, under certain hypotheses on $h$, "most" Dehn-fillings of $M$ (in some cases all but finitely many) are virtually $\mathbb{Z}$-representable. We apply our results to show that surgeries on the figure-eight knot with even numerator are virtually $\mathbb{Z}$-representable.

Generalized torsions in once punctured torus bundles

A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without generalized torsions, it is conjectured that 3-manifold groups without generalized torsions are bi-orderable. In this paper, we find generalized torsions in the fundamental groups of once punctured torus bundles which are not bi-orderable. Our result contains a generalized torsion in a tunnel number two hyperbolic once punctured torus bundle.

Guts of surfaces in punctured-torus bundles

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.

Nonorientable, incompressible surfaces in punctured-torus bundles over S^1

We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to the projection of the bundle into the basis S^1.

Branched Cauchy-Riemann Structures on Once-Punctured Torus Bundles

Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of $3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and…

Self-mapping degrees of torus bundles and torus semi-bundles

Each closed oriented 3-manifold $M$ is naturally associated with a set of integers $D(M)$, the degrees of all self-maps on $M$. $D(M)$ is determined for each torus bundle and torus semi-bundle $M$. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine $D(M)$ for all 3-manifolds in Thurston's picture.

The canonical decomposition of once-punctured torus bundles

In this paper, we determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defined by Floyd and Hatcher. Unlike previous work on this problem, the techniques we use are not geometric. Instead, they involve angled polyhedral decompositions, thin position and a version of almost normal surface theory.

Complexity of torus bundles over the circle with monodromy (2 1, 1 1)

We find the exact values of complexity for an infinite series of 3-manifolds. Namely, by calculating hyperbolic volumes, we show that c(N_n)=2n, where $c$ is the complexity of a 3-manifold and N_n is the total space of the punctured torus bundle over S^1 with monodromy 2&1 1&1 ^n$. We also apply a recent result of Matveev and Pervova to show that c(M_n) \ge 2Cn with C\approx 0.598, where a compact manifold M_n is the total space of the torus bundle over S^1 with the same monodromy as N_n, and discuss an approach to the conjecture c(M_n)=2n+5 based on the equality c(N_n)=2n.

界定有理同调圆的环面丛的分类

Classification of torus bundles that bound rational homology circles

本文完全分类了界定具有圆的有理同调的4-流形的圆上环面丛，并探讨了链环整数手术及3-辫闭合双重分支覆盖与有理同调球的联系。

Sol流形中的Z₂-Thurston规范与非定向曲面的可嵌入性

$\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces

本文研究Sol流形中Z₂-Thurston规范的计算方法，并确定非定向闭曲面在其中的可嵌入性条件。

Virtually Haken surgeries on once-punctured torus bundles

We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of ``most'' commensurability classes of punctured torus bundles. In particular, we define an integer-valued complexity function on monodromies $f$ (essentially the length of the LR-factorization of $f_*$ in $PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$ has co…

Detection of incompressible surfaces in hyperbolic punctured torus bundles

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We convert non fiber incompressible surfaces from their form to the form output by Yoshida's construction, and run his construction backwards to give (for non semi-fibers, which we identify) the data needed to construct ideal points of the deformation variety corresponding to those surfac…

Monodromy of torus fiber bundles and decomposability problem

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.

On the topology of the space of contact structures on torus bundles

We prove the existence of essential loops in the space of contact structures on torus bundles over the circle.

On symplectic fillings of virtually overtwisted torus bundles

We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1-handle attachment, and we give a condition under which distinct tight lens space filling…

关于一次 punctured torus 系列的特征品种拓扑

On the topology of character varieties of once-punctured torus bundles

本文研究了一次 punctured torus 系列的特征品种拓扑，通过分析纤维上的特征并构建分支覆盖映射，扩展了 Boyer 等人的成果，并展示了无限家族的超几何 bundles 的 canonical curves 特征具有无界 genus。

Elliptic open books on torus bundles over the circle

As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.

Strong symplectic fillability of contact torus bundles

In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually…

On Liftings of Local Torus Actions to Fiber Bundles

In this note we define a lifting of a local torus action modeled on the standard representation (we call it a local torus action for simplicity) to a principal torus bundle, and show that there is an obstruction class for the existence of liftings in the first cohomology of the fundamental group of the orbit space with coefficients in a certain module.

Symplectic fillability of tight contact structures on torus bundles

We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.

Fintushel--Stern knot surgery in torus bundles

Suppose that $X$ is a torus bundle over a closed surface with homologically essential fibers. Let $X_K$ be the manifold obtained by Fintushel--Stern knot surgery on a fiber using a knot $K\subset S^3$. We prove that $X_K$ has a symplectic structure if and only if $K$ is a fibered knot. The proof uses Seiberg--Witten theory and a result of Friedl--Vidussi on twisted Alexander polynomials.

Local geometry of symplectic divisors with applications to contact torus bundles

In this note we study the contact geometry of symplectic divisors. We show the contact structure induced on the boundary of a divisor neighborhood is invariant under toric and interior blow-ups and blow-downs. We also construct an open book decomposition on the boundary of a concave divisor neighborhood and apply it to the study of universally tight contact structures of contact torus bundles.

通过扭曲Alexander多项式的单穿孔环面丛的射影刚性

Projective Rigidity of Once-Punctured Torus Bundles via the Twisted Alexander Polynomial

本文通过扭曲Alexander多项式验证双曲单穿孔环面丛的射影刚性，并将其与字符簇切空间上的诱导作用及谱序列中的群论作用联系起来。

Character varieties of once-punctured torus bundles with tunnel number one

We determine the PSL_2(C) and SL_2(C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine `natural' models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Ad…

同伦等价于透镜空间上某些环面丛的流形

Manifolds homotopy equivalent to certain torus bundles over lens spaces

本文首次计算了一类基本群不能由无挠群和有限群经融合积与 HNN 扩张得到的闭拓扑流形的拓扑简单结构集，并利用分裂阻碍和 rho 不变量等经典手术不变量刻画简单同伦等价何时同伦于同胚。

穿孔环面丛上的非标准双向序

Non-standard bi-orders on punctured torus bundles

本文研究穿孔环面丛群的非标准双向序，证明其最大及次大真凸子群与标准双向序一致，并确认这些群确实存在非标准双向序。

Azumaya algebras and once-punctured torus bundles

Recent work of Chinburg, Reid, and Stover has shown that certain arithmetic and algebro-geometric properties of the character variety of a hyperbolic knot complement in the 3-sphere $M=S^3\setminus K$ yields topological and number theoretic information about Dehn fillings of M. Specifically, they show how the study of a certain extension problem for quaternion Azumaya algebras is related to topological invariants associated to these fillings. In this paper, we extend their work to the setting of hyperbolic once punctured torus bundles. Along the way, we exhibit new phenomenon in the relevant e…

Torus bundles not distinguished by TQFT invariants

We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in $SL(2,\Z)$ and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and lastly by Ng and Schauenburg for the Drinfeld doubles of spherical fusi…

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, $T_ϕ$, over the circle with pseudo-Anosov monodromy $ϕ$, there are associated two tessellations of the complex plane: one, $Δ(ϕ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(ϕ)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $Δ(ϕ)$ and $CW(ϕ)$.

Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Knots

We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.

用特定 Morse-Bott 函数刻画表示为 Lens 空间、$S^2 \ imes S^1$ 和环面在圆上的丛的连通和的 $3$ 维流形

Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions

本文用特定 Morse-Bott 函数刻画表示为 Lens 空间、$S^2 \ imes S^1$ 和环面在圆上的丛的连通和的 $3$ 维流形，强化并明确了 Saeki 2006 年的相关结果。

Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

Given a matrix $A\in SL(N,\Z)$, form the semidirect product $G=\Z^N\rtimes_A \Z$ where the $\Z$ factor acts on $\Z^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this paper we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.

经典和量子算法用于紧致纤维丛的拓扑不变量

Classical and Quantum Algorithms for Topological Invariants of Torus Bundles

本文提出经典和量子算法计算紧致纤维丛的拓扑不变量，通过非交换环面结构实现高效计算，展现量子算法在空间复杂度上的优势。

透镜空间上的环面丛

Torus bundles over lens spaces

本文推广Davis和Lück关于透镜空间上环面丛的L-群和结构集的计算，将其覆盖到$\mathbb{Z}/p$在$\mathbb{Z}^n$上的所有作用，包括奇异集非离散的情形。

Fixed Points and Coincidences in Torus Bundles

Minimum numbers of fixed points or of coincidence components (realized by maps in given homotopy classes) are the principal objects of study in topological fixed point and coincidence theory. In this paper we investigate fiberwise analoga and represent a general approach e.g. to the question when two maps can be deformed until they are coincidence free. Our method involves normal bordism theory, a certain pathspace EB and a natural generalization of Nielsen numbers. As an illustration we determine the minimum numbers for all maps between torus bundles of arbitrary (possibly different) dimensio…

主环面丛拓扑 T-对偶性的一种新方法

A new approach to topological T-duality for principal torus bundles

本文提出用“Thom 类”重新表述主环面丛的拓扑 T-对偶性，在保持与既有定义等价的同时取消 H-通量的全局假设，并简化分类证明、推广圆周丛的局部理论。

Torus bundles and 2-forms on the universal family of Riemann surfaces

We revisit three results due to Morita expressing certain natural integral cohomology classes on the universal family of Riemann surfaces C_g, coming from the parallel symplectic form on the universal jacobian, in terms of the Miller-Morita-Mumford classes e and e_1. Our discussion will be on the level of the natural 2-forms representing the relevant cohomology classes, and involves a comparison with other natural 2-forms representing e, e_1 induced by the Arakelov metric on the relative tangent bundle of C_g over M_g. A secondary object called a_g occurs, which was discovered and studied by K…

自由环面作用与扭曲悬挂

Free torus actions and twisted suspensions

本文将两流形连通和上的主圆丛全空间表示为各加项上的圆丛全空间，以此为球面乘积连通和上的自由圆与环面作用提供存在充分条件，并对具自由余齐次四环面作用的闭单连通流形进行拓扑分类，推得无限多具正 Ricci 曲率与等距环面作用的流形。

阿廷辫群的晶体商与平坦$3$流形上的环面丛

Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds

本研究针对阿廷辫群的晶体商，证明紧致连通平坦流形可作为几何环面丛的基，确定相关环面丛群扩张的分裂情况，明确四类紧致平坦流形的分裂情形。

闭合定向曲面上的可定向环面束的分类

Classification of orientable torus bundles over closed orientable surfaces

本文研究了闭合定向曲面上的可定向环面束的分类问题，通过分析映射类群作用下的同态分类，给出了束同构下的完整分类结果，并探讨了其在拓扑扩展问题中的应用。

Real Places and Surface Bundles

Our results complement D. Calegari's result that there are no hyperbolic once-punctured torus bundles over $S^1$ with trace field having real place. We exhibit several infinite families of pairs $(-χ, p)$ such that there exist hyperbolic surface bundles with over $S^1$ with fiber having $p$ punctures and Euler characteristic $χ$ with trace field of having a real place. This supports our conjecture that there exist such examples for each pair $(-χ, p)$ with finitely many known exceptions.

From Torus Bundles to Particle-Hole Equivariantization

We continue the program of constructing (pre)modular tensor categories from 3-manifolds first initiated by Cho-Gang-Kim using $M$ theory in physics and then mathematically studied by Cui-Qiu-Wang. An important structure involved is a collection of certain $\text{SL}(2, \mathbb{C})$ characters on a given manifold which serve as the simple object types in the corresponding category. Chern-Simons invariants and adjoint Reidemeister torsions play a key role in the construction, and they are related to topological twists and quantum dimensions, respectively, of simple objects. The modular $S$-matri…