三维流形的挠

The L^2-Alexander torsion of 3-manifolds

We introduce $L^2$-Alexander torsions for 3-manifolds, which can be viewed as a generalization of the $L^2$-Alexander polynomial of Li--Zhang. We state the $L^2$-Alexander torsions for graph manifolds and we partially compute them for fibered manifolds. We furthermore show that given any irreducible 3-manifold there exists a coefficient system such that the corresponding $L^2$-torsion detects the Thurston norm.

Exponential Torsion Growth for Random 3-Manifolds

We show that a random 3-manifold with positive first Betti number admits a tower of cyclic covers with exponential torsion growth.

Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel-Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

Twisted Reidemeister torsion and the Thurston norm: graph manifolds and finite representations

We show that the Thurston norm of any irreducible 3-manifold can be detected using twisted Reidemeister torsions corresponding to integral representations and also corresponding to representations over finite fields. In particular our result holds for all graph manifolds, these are not covered by the earlier work of the first author and Vidussi.

具有指数同调 torsion 增长的双曲流形族

Family of hyperbolic manifolds with exponential homology torsion growth

本文构造了双曲流形族，展示同调 torsion 的指数增长，证明最近关于同调 torsion 的界是渐近最优的。

Twisted Reidemeister torsion, the Thurston norm and fibered manifolds

We prove that the twisted Reidemeister torsion of a 3-manifold corresponding to a fibered class is monic and we show that it gives lower bounds on the Thurston norm. The former fixes a flawed proof in [FV10], the latter gives a quick alternative argument for the main theorem of [FK06].

具有Legendrian链和边界模式的Combed三维流形的挠不变量

Torsion Invariants of Combed 3-Manifolds with Boundary Pattern and Legendrian Links

我们通过允许（合适的）边界相切圆和非零欧拉示性数的流形，扩展了Turaev关于配备非奇异向量场的三维流形的挠不变量的定义。我们表明，这些不变量特别适用于（外部）Legendrian联系3-流形。我们的方法使用组合编码的向量场，基于标准的脊椎。本文将这种编码从闭流形推广到有边流形。

三维流形的挠率和Seiberg-Witten不变量的Surgery公式

Surgery formula for torsions and Seiberg-Witten invariants of 3-manifolds

给出了三维流形上$Spin^c$-结构的挠率和Seiberg-Witten不变量的一个外科手术公式。我们使用的技术Reidemeister型扭转和他们的改进。

Reidemeister torsion of a 3-manifold obtained by a Dehn-surgery along the figure-eight knot

Let M be a closed 3-manifold obtained by Dehn surgery along the figure-eight knot. We give a formula of the Reisdemeisiter torsion of M for any irreducible SL(2;C)-representation. It is described as a rational expression of the trace of the image of the meridian.

Hidden torsion, 3-manifolds, and homology cobordism

This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifolds which have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger-Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pairwise homology cobordant, yet remain indistingui…

Reidemeister torsion, hyperbolic three-manifolds, and character varieties

This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume. Torsions are viewed as topological invariants and also as functions on the variety of representations in $\operatorname{ SL}_2(\mathbb C)$. In both cases, the torsions may also be computed after composing with finite dimensional representations of $\operatorname{ SL}_2(\mathbb C)$. In addition the paper deals with the torsion of the adjoint representation as a function on the variety of $\operatorname{ PSL}_{n+1}(\mathbb C)$-characters, using that the first cohomology group with coefficients twisted by th…

沿某些双扭结做0-手术得到的3-流形基本群中的广义挠元

Generalized torsion elements in the fundamental groups of 3-manifolds obtained by the 0-surgeries along some double twist knots

该研究针对沿双扭结做0-手术得到的3-流形，构造其基本群中广义挠元的候选元并验证存在性，借助JSJ分解完成证明，同时得出该类3-流形基本群可双序等价于无广义挠元的结论，还列举了相关实例。

三维流形的Reidemeister-Turaev挠率的有限性

Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds

证明了闭定向三流形的Reidemeister-Turaev挠在Goussarov和Habiro意义下的有限性，即关于保持流形同调型的剪切粘贴操作的有限性.一般来说，这些性质要求流形配备欧拉结构和同调参数化。

An introduction to the abelian Reidemeister torsion of three-dimensional manifolds

These notes accompany some lectures given at the autumn school "Tresses in Pau" in October 2009. The abelian Reidemeister torsion for 3-manifolds, and its refinements by Turaev, are introduced. Some applications, including relations between the Reidemeister torsion and other classical invariants, are surveyed.

A remark on torsion growth in homology and volume of 3-manifolds

We show that Lück's conjecture on torsion growth in homology implies that two 3-manifolds have equal volume if the fundamental groups have the same set of finite quotients.

Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL_2(C) to the corresponding Reidemeister torsion. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.

Torsion in the homology of finite covers of 3-manifolds

Let $N$ be a prime 3-manifold that is not a closed graph manifold. Building on a result of Hongbin Sun and using a result of Asaf Hadari we show that for every $k\in\Bbb{N}$ there exists a finite cover $\tilde{N}$ of $N$ such that $|\operatorname{Tor} H_1(\tilde{N};\Bbb{Z})|>k$.

Asymptotics of analytic torsion for hyperbolic three--manifolds

We prove that for certain sequences of hyperbolic three--manifolds with cusps which converge to hyperbolic three--space in a weak ("Benjamini-Schramm") sense and certain coefficient systems the regularized analytic torsion approximates the $L^2$-torsion of the universal cover under an additional hypothesis. We also prove an asymptotic equality between the former and some Reidemeister torsion of the truncated manifolds.

Counting non-commensurable hyperbolic manifolds and a bound on homological torsion

We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp. We also determine how the number of non-commensurable closed hyperbolic manifolds of dimension at least 3 and bounded diameter grows. The lower bound implies that the fraction of arithmetic manifolds tends to zero as the diameter goes up.

Generalized torsion and decomposition of 3-manifolds

A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable $3$-manifold $M$ has a generalized torsion element if and only if the fundamental group of some prime factor of $M$ has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal $3$-manifolds whose fund…

The asymptotic behavior of the Reidemeister torsion for Seifert manifolds and PSL(2;R)-representations of Fuchsian groups

We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PSL(2;R)-representation of a Fuchsian group. In particular, the leading coefficient of the Reidemeister torsion for the unit tangent bundle over a two-orbifold converges to $-χ\log2$ where $χ$ is the Euler characteristic of the two-orbifold. We also give a relation between $\mathbb…

Turaev torsion, definite 4-manifolds, and quasi-alternating knots

We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative definite 4-manifold with trivial first homology and bounded second betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.

论3维流形的扭曲 $L^2$-挠率的正性

On the positivity of twisted $L^2$-torsion for 3-manifolds

本文研究带空或不可压缩环面边界的紧致可定向不可约3维流形上的扭曲 $L^2$-挠率，证明无限基本群情形下其严格为正，并在上三角表示子簇上连续。

$3$-流形的$\mathfrak{gl}(1 \vert 1)$-Alexander多项式、Reidemeister挠率和透镜空间

$\mathfrak{gl}(1 \vert 1)$-Alexander polynomial for $3$-manifolds, Reidemeister torsion and lens spaces

本文重新表述了先前定义的不变量$Δ(M, ω)$，揭示了其与Reidemeister挠率的关系，并在$3$-流形为透镜空间时计算了该不变量，进而探讨了利用它对透镜空间的分类。

Generalized torsion elements and bi-orderability of 3-manifold groups

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite families of closed hyperbolic 3-manifolds. In the course of the proof, we prove that each standard generator of the Fibonacci group F(2,m) (m>2) is a generalized torsion element.

Torsion homology growth and cycle complexity of arithmetic manifolds

Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of `low' genus, and give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.

Universal $L^2$-torsion, polytopes and applications to $3$-manifolds

Given an $L^2$-acyclic connected finite $CW$-complex, we define its universal $L^2$-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group $\operatorname{Wh}^w(G)$. We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group $\operatorname{Wh}^w(G)$ to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal $L^2$-torsion can be identifie…

Virtual Homological Torsion of Closed Hyperbolic 3-manifolds

In this paper, we will use Kahn-Markovic's almost totally geodesic surfaces to construct certain $π_1$-injective 2-complexes in closed hyperbolic 3-manifolds. Such 2-complexes are locally almost totally geodesic except along a 1-dimensional subcomplex. Using Agol and Wise's result that fundamental groups of hyperbolic 3-manifolds are LERF and quasi-convex subgroups are virtual retract, we will show that closed hyperbolic 3-manifolds virtually contain any prescribed homological torsion: For any finite abelian group $A$, and any closed hyperbolic 3-manifold $M$, there exists a finite cover $N$ o…

Survey on analytic and topological torsion

The article consists of a survey on analytic and topological torsion. Analytic torsion is defined in terms of the spectrum of the analytic Laplace operator on a Riemannian manifold, whereas topological torsion is defined in terms of a triangulation. The celebrated theorem of Cheeger and Müller identifies these two notions for closed Riemannian manifolds. We also deal with manifolds with boundary and with isometric actions of finite groups. The basic theme is to extract topological invariants from the spectrum of the analytic Laplace operator on a Riemannian manifold.

Algebraic Torsion in Contact Manifolds

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of…

Torsion and symplectic volume in Seifert manifolds

For any oriented Seifert manifold X and compact connected Lie group G with finite center, we relate the Reidemeister density of the moduli space of representations of the fundamental group of X into G to the Liouville measure of some moduli spaces of representations of surface groups into G.

Prescribed virtual homological torsion of 3-manifolds

We prove that given any finite abelian group $A$ and any irreducible $3$-manifold $M$ with empty or toroidal boundary which is not a graph manifold there exists a finite cover $M' \to M$ so that $A$ is a direct factor in $H_1(M',\mathbb{Z})$. This generalizes results of Sun and of Friedl-Herrmann.

Analytic torsion and R-torsion of Witt representations on manifolds with cusps

We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a cl…

Twisted torsion invariants and link concordance

The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.

Centralizers of torsion in 3-manifold groups

We show that if $P$ is a $PD_3$-complex and $g\inπ_1(P)$ has finite order $>1$ and infinite centraliser then $π_1(P)$ retracts onto $Z/2Z\oplus\mathbb{Z}$. If $P$ is an irreducible closed 3-manifold then it follows from the Projective Plane Theorem that $P\cong{RP^2}\times{S^1}$.

Exponential growth of torsion in the cohomology of arithmetic hyperbolic manifolds

For d=2n+1 a positive odd integer, we consider sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding corresponding hyperbolic manifolds of finite volume and show that, under appropriate and natural assumptions, the torsion of the associated cohomology groups grows exponentially.

3维流形的扩张性与挠同调

Expansion and torsion homology of 3-manifolds

本文构造在所有维度均为良好有理扩张子的3维流形，证明扩张子必有大量挠同调，并给出拓扑重叠问题中具有大宽度的例子。

高维接触流形中的代数挠率与 Giroux 挠率

Algebraic and Giroux torsion in higher-dimensional contact manifolds

本文在任意奇数维构造了具有有限非零代数挠率的接触流形，证明其 tight 且无强辛填充，并证明 Giroux 挠率蕴含代数 1-挠率，从而解决 Massot-Niederkrueger-Wendl 猜想。

Twisted Alexander invariant and non-abelian Reidemeister torsion for hyperbolic three-dimensional manifolds with cusps

We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative of the twisted Alexander invariant for a hyperbolic knot exterior gives the hyperbolic torsion. We prove such a derivative formula of the twisted Alexander invariant for hyperbolic link exteriors like the Whitehead link exterior. We provide the framework for the derivative formula to work, which consists of assumptions on the topology of the manifold and on t…

The L^2-torsion function and the Thurston norm of 3-manifolds

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element ϕin the first cohomology of M with integral coefficients. Then one can define the ϕ-twisted L^2-torsion function of the universal covering which is a function from the set of positive real numbers to the set of real numbers. By earlier work of the second author and Schick the evaluation at t=1 determines the volume. In this paper we show that its degree, which is a number extracted from its asymptotic behavior at 0 and at infinity, agrees with the Thurston norm of ϕ.

On the classification of 1-connected 7-manifolds with torsion free second homology

We generalize a result of the author about the classification of 1-connected 7-manifolds and demonstrate its use by two concrete applications, one to 2-connected 7-manifolds (a new proof -- and slightly different formulation -- of an up to now unpublished Theorem by Crowley and Nordstroem and one to simply connected 7-manifolds with the cohomology ring of $S^2 \times S^5 \sharp S^3 \times S^4$. The answer is in terms of generalized Kreck-Stolz invariants, which in the case of 2-connected 7-manifolds is equivalent to a quadratic refinement of the linking form and a generalized Eells-Kuiper inva…

通用 $L^2$- torsion 和 3-流形的缝合分解

Universal $L^2$-torsion and sutured decomposition for 3-manifolds

本文研究了 3-流形的通用 $L^2$-torsion 及其在缝合分解中的应用，证明了该不变量能检测纤维性并推导了分解公式。

Generalized torsion for hyperbolic $3$--manifold groups with arbitrary large rank

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the trivial element, then $g$ is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic $3$--manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer $n > 1$ there are infinitely many closed hyperbolic $3$--manifolds $M_n$ which enjoy the property: (i) the Heegaard genus of $M_n$ is $n$, (ii) the rank of the fundamental grou…

纤维化3-流形的1-loop等于挠率

1-loop equals torsion for fibered 3-manifolds

本文证明了纤维化3-流形的两个1-loop多项式不变量分别等于$\mathbb{C}^2$和$\mathbb{C}^3$挠率多项式，并在超过6,600个非纤维化流形上验证了该猜想，同时探讨了$\mathbb{C}^2$挠率与Thurston范数的关系。

Resolvent, heat kernel and torsion under degeneration to fibered cusps

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.

3-流形的超表示与挠多项式

Super-representations of 3-manifolds and torsion polynomials

本文通过引入涉及边和面的新组合结构，将多项式关联到尖点双曲流形，猜想其与$\BC^2$-挠多项式一致，从而连接纽结亏格与几何表示，并受超几何启发。

Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary

In this paper, we define a relative $L^2$-$ρ$-invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of positive scalar curvature metrics which are collared near the boundary. As an application, we show that if a $4k+3$-dimensional spin manifold with boundary admits such a metric and if, roughly speaking, there exists a torsion element in the difference of the fundamental groups of the manifold and its boundary, then there are infinitely many bordism classes of such psc metrics on the given manifold. This result in…

On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic 3-manifold and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the c…

三维流形群中二阶广义挠元素的分类

Classification of generalized torsion elements of order two in 3-manifold groups

本文分类了基本群含二阶广义挠元素的三维流形及其中的相应元素，并证明三维流形群的 R-群与 R-bar 群重合，给出完整分类。

接触Ozsvath-Szabo不变量与Giroux挠率

Contact Ozsvath-Szabo Invariants and Giroux Torsion

本文证明了具有正Giroux挠率的切触3-流形的切触Ozsvath-Szabo不变量的一个消灭定理。我们使用这个结果建立类似的消失结果切触结构与基础3-流形承认无论是环面纤维化的圆或塞弗特纤维化的定向基地。我们还表明-使用接触拓扑的标准技术-如果接触3-流形（Y，）有正的Giroux挠，那么存在一个Stein协边从（Y，）接触3-流形（Y，'），使（Y，）是从（Y，'）通过Lutz修改。

通过 Dehn 手术得到的某些 3-流形的伴随 Reidemeister 挠率

Adjoint Reidemeister torsions of some 3-manifolds obtained by Dehn surgeries

本文计算沿八字结或 5_2 结做 Dehn 手术所得 3-流形的伴随 Reidemeister 挠率，并证明相关类似消失猜想成立。