首页/论文/1609.07809

arXiv 1609.078092016-09math.GT

泛$L^2$ -挠、多胞形及其对$3$ -流形的应用

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

Stefan Friedl, Wolfgang Lück

备注：: 37 pages, final version, to be published by Proc. London Math. Soc
DOI：: 10.1112/plms.12035

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

摘要英文摘要相关论文

摘要：

给定一个$L^2$ -非循环连通有限$CW$ -复形，我们根据其泛覆盖的链复形定义其泛$L^2$ -挠。它取弱Whitehead群$\operatorname{Wh}^w(G)$中的值。研究了它的同伦不变性、求和公式、乘积公式和Poincaré对偶等主要性质。在一定的假设下，我们可以指定从弱Whitehead群$\operatorname{Wh}^w(G)$到阿贝尔群（如实数或整多面体的Grothendieck群）的某些同态，并且通用的$L^2$ -扭转的图像可以用许多不变量来识别，例如$L^2$ -扭转，$L^2$ -扭转函数，扭转$L^2$ -欧拉示性数，在$3$歧管的情况下，对偶Thurston范数多面体

英文摘要：

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 identified with many invariants such as the $L^2$-torsion, the $L^2$-torsion function, twisted $L^2$-Euler characteristics and, in the case of a $3$-manifold, the dual Thurston norm polytope.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

摘要英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

$\Bbb C^n$、复流形和凸多胞体中的实二次曲面

Real quadrics in $\Bbb C^n$, complex manifolds and convex polytopes

本文研究了一类非Kähler紧复流形的拓扑结构，推广了Hopf和Calabi-Eckmann流形的拓扑结构。这些流形与$\Bbb C^n$中的实二次曲面的特殊系统是同构的，这些系统对于实环面$(\Bbb S^1)^n$到$\Bbb C^n$上的自然作用是不变的。商空间是一个简单的凸多胞体。这个问题减少，从而研究的拓扑结构的某些真正的代数集，并可以处理使用组合的结果对凸多胞体。我们证明了这些紧致复流形的同调群可以具有任意的挠率，从而使得它们的拓扑极其丰富。通过引入与简单凸多胞体的某些变换相关的全纯等变初等运算，我们还解决了一个相关的跨壁问题。最后，作为一个很好的结果，我们得到了仿射非Kähler紧复流形在其同调群中可以有任意量的挠率，与Kähler情形相反.

2004-05-05math.GT

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。

2025-12-01math.GT

Thurston 范数、多面体与分裂复杂度

Thurston norm, polytopes and splitting complexity

Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM（西班牙高等科学研究理事会-马德里自治大学-马德里理工大学-马德里康普顿斯大学联合数学科学研究所）；Emmanuel College, University of Cambridge（剑桥大学伊曼纽尔学院）

研究满足强Atiyah猜想的无挠群，将核的第一L²-Betti数延拓为Thurston半范数，证明其由同调群中的多面体诱导，并推广到高阶L²-Betti数；在自由-循环群和三维流形基本群情形给出组合解释，应用于计算BNS不变量。

2026-09-02math.GRmath.GT

关于给定直径的四维集合的泛覆盖

On universal covers for four-dimensional sets of a given diameter

Makeev证明，在中心对称的四维多面体，有超过20个方面和外切的欧几里德球直径1，有没有普遍覆盖家庭的单位直径集。本文研究了这一逆问题，证明了每一个最多有14个刻面且外切于直径为1的欧氏球的中心对称多面体是单位直径集族的泛覆盖。

2010-10-22math.GTmath.MG

Seifert surgery on knots via Reidemeister torsion and Casson-Walker-Lescop invariant III

For a knot $K$ in a homology $3$-sphere $Σ$, let $M$ be the result of $2/q$-surgery on $K$, and let $X$ be the universal abelian covering of $M$. Our first theorem is that if the first homology of $X$ is finite cyclic and $M$ is a Seifert fibered space with $N\ge 3$ singular fibers, then $N\ge 4$ if and only if the first homology of the universal abelian covering of $X$ is infinite. Our second theorem is that under an appropriate assumption on the Alexander polynomial of $K$, if $M$ is a Seifert fibered space, then $q=\pm 1$ (i.e.\ integral surgery).

2018-03-16math.GT

The Bieri-Neumann-Strebel invariants via Newton polytopes

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS)…

2019-09-03math.GRmath.GTmath.RA

更多相关论文收起更多论文

Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups

We investigate Friedl-Lück's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb{Z}$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the unive…

2018-01-29math.GRmath.GT

具有有限维表示的扭转$L^2$ -不变量

Twisting $L^2$-invariants with finite-dimensional representations

我们研究如何用有限维表示扭曲L^2-不变量，如L^2-贝蒂数和L^2-挠。作为一种特殊情况，我们给有限连通CW-复形X的泛覆盖加上一个从R^{>0}到R的φ-扭曲L^2-挠函数，条件是X的基本群是剩余有限的，并且它的泛覆盖是L^2-无圈的。

2017-03-27math.GT

三维流形的L^2-扭函数与Thurston范数

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

设M是具有无限基本群和空边界或环形边界的定向不可约3-流形。考虑M的第一上同调中的任意元素n，其系数为整。然后可以定义泛覆盖的n-扭曲L^2-扭函数，它是从正实数集到实数集的函数。根据第二作者和Schick的早期工作，t=1时的估值决定了体积。在本文中，我们表明，它的程度，这是一个数字提取的渐近行为在0和无穷大，同意的Thurston范数。

2015-11-18math.GT

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.

2018-02-13math.DGmath.GT

群复形的挠不变量

Torsion invariants of complexes of groups

本文研究剩余有限群作用在可缩复形上的同调挠增长，证明归一化挠极限等于基本域边界子复形的挠（度数平移一位），并给出直角 Artin 群的完整答案。

2024-01-17math.GRmath.GT

两个双曲5-流形的整同余

Integral Congruence Two Hyperbolic 5-Manifolds

本文对所有由双曲空间形式$H^5/Γ$产生的可定向双曲5-流形进行了分类，其中$Γ$是Lorentz二次型$x_1^2+...+x_5^2-x_6^2$的正单位群$Γ^5$的同余二子群$Γ^5_2$的最小指数的无挠子群.我们还证明了$Γ^5_2$是关于$H^5$中的5维直角凸多胞体的反射群。作为应用，我们构造了一个已知体积最小的双曲5-流形$7ζ(3)/4$ .

2003-08-13math.DGmath.GT

Residually finite lattices in $\widetilde{\mathrm{PU}(2,1)}$ and fundamental groups of smooth projective surfaces

This paper studies residual finiteness of lattices in the universal cover of $\mathrm{PU}(2,1)$ and applications to the existence of smooth projective varieties with fundamental group a cocompact lattice in $\mathrm{PU}(2,1)$ or a finite covering of it. First, we prove that certain lattices in the universal cover of $\mathrm{PU}(2,1)$ are residually finite. To our knowledge, these are the first such examples. We then use residually finite central extensions of torsion-free lattices in $\mathrm{PU}(2,1)$ to construct smooth projective surfaces that are not birationally equivalent to a smooth co…

2021-12-30math.AGmath.GT

映射类群阈值处的挠元

Torsion at the Threshold for Mapping Class Groups

本文研究带标记点的闭可定向曲面映射类群的离散泛欧拉类幂次$n=g$的情形，证明该幂次对应的类为挠类，其生成的循环子群的阶是$4g(2g+1)(2g-1)$的正整数倍。

2024-09-11math.ATmath.GT