Geometric Topology

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New submissions (showing 5 of 5 entries )

[1] arXiv:2509.20685 [cn-pdf, pdf, html, other]

Title: A Morse complex for the homology of vanishing cycles Show Chinese title

Title: 一个关于消失循环同调的莫尔斯复形

Aleksander Doan, Juan Muñoz-Echániz

Comments: 29 pages. Comments are welcome!

Subjects: Geometric Topology (math.GT) ; Algebraic Geometry (math.AG) ; Differential Geometry (math.DG)

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of the function, built from a normal crossing compactification of the variety. They are canonically isomorphic to the homology of vanishing cycles and -- in the absence of bifurcations at infinity -- recover the hypercohomology of the perverse sheaf of vanishing cycles, studied extensively in singularity theory and enumerative geometry. Our construction arises as a special case of a more general construction of Morse homology of non-compact manifolds that admit a compactification by a manifold with corners. Show Chinese abstract

我们构造与光滑复代数簇上任何正则函数相关的莫尔斯同调群，允许奇点和非紧致临界点集。 这些群由该函数的某种大扰动的临界点生成，该扰动从簇的正常相交紧化构建而来。 它们与消失循环的同调群典范同构，并且——在无穷远处没有分支的情况下——恢复了消失循环的 perverse 穿透层的上同调，这在奇点理论和枚举几何中得到了广泛研究。 我们的构造是更一般的非紧流形的莫尔斯同调构造的一个特例，这些流形可以通过一个有角的流形进行紧化。

[2] arXiv:2509.21069 [cn-pdf, pdf, html, other]

Title: Lefschetz pencils on a complex projective plane from a topological viewpoint Show Chinese title

Title: 从拓扑观点看复射影平面上的Lefschetz铅笔

Ju A Lee

Comments: 29 pages, 23 Figures

Subjects: Geometric Topology (math.GT)

In this article, we present a differential topological construction of symplectic Lefschetz pencils of genus $\frac{(d-1)(d-2)}{2}$ with $d^2$ base points and $3(d-1)^2$ critical points for arbitrary $d\geq 4$, analogous to the holomorphic Lefschetz pencils of curves of degree $d$ in $\mathbb{C}P^2$. Moreover, for the case $d=4$, we derive an explicit monodromy factorization of the genus $3$ holomorphic Lefschetz pencil on $\mathbb{C}P^2$ based on the braid monodromy technique and prove that it can also be topologically constructed by breeding the monodromy relations of the genus $1$ holomorphic Lefschetz pencils. Show Chinese abstract

在本文中，我们提出了一种微分拓扑构造，用于生成具有$\frac{(d-1)(d-2)}{2}$个亏格、$d^2$个基点和$3(d-1)^2$个临界点的辛Lefschetz铅笔，对于任意的$d\geq 4$，类似于次数为$d$的曲线在$\mathbb{C}P^2$中的全纯Lefschetz铅笔。 此外，对于情况$d=4$，我们基于辫子单值性技术推导出在$\mathbb{C}P^2$上的 genus$3$全纯 Lefschetz 线丛的显式单值分解，并证明它也可以通过 genus$1$全纯 Lefschetz 线丛的单值关系进行拓扑构造。

[3] arXiv:2509.21093 [cn-pdf, pdf, html, other]

Title: The Euler class of infinite-type surface bundles Show Chinese title

Title: 无限类型曲面丛的欧拉类

Mauricio Bustamante, Rita Jiménez Rolland, Israel Morales

Comments: 17 pages

Subjects: Geometric Topology (math.GT) ; Algebraic Topology (math.AT)

We study the Euler class of smooth orientable infinite-type surface bundles with a section. For many such surfaces, we show that this cohomology class is nontrivial, and that the behavior of its powers depends on the genus and the type of ends. As an application, we extend Morita's non-lifting theorem to many infinite-type surfaces, including surfaces of infinite genus. Show Chinese abstract

我们研究具有截面的光滑定向无限类型曲面丛的欧拉类。 对于许多这样的曲面，我们证明这个上同调类是非平凡的，并且其幂的行为取决于亏格和端点的类型。 作为应用，我们将莫里塔的非提升定理扩展到许多无限类型曲面，包括无限亏格的曲面。

[4] arXiv:2509.21140 [cn-pdf, pdf, html, other]

Title: Every negative amphichiral knot is rationally slice Show Chinese title

Title: 每个负的互反扭结都是有理分片

Alessio Di Prisa, Jaewon Lee, Oğuz Şavk

Comments: 36 pages, 23 figures. Comments are welcome!

Subjects: Geometric Topology (math.GT)

In 2009, Kawauchi proved that every strongly negative amphichiral knot is rationally slice. However, as shown by Hartley in 1980, there are examples of negative amphichiral knots that are not strongly negative amphichiral. In this paper, we prove that every negative amphichiral link whose amphichiral map preserves each component is rationally slice. Our proof relies on a systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior. Moreover, we provide sufficient conditions on such an action to deduce when a negative amphichiral knot is either isotopic to, or concordant to, a strongly negative amphichiral knot. In particular, we prove that every fibered negative amphichiral knot is strongly negative amphichiral, answering a question asked by Kim and Wu in 2016 on Miyazaki knots. Show Chinese abstract

2009年，Kawauchi证明了每个强负互为镜像的扭结都是有理截断的。 然而，如Hartley在1980年所示，存在一些负互为镜像的扭结不是强负互为镜像的。 在本文中，我们证明了每个互为镜像映射保持每个分量的负互为镜像链环都是有理截断的。 我们的证明依赖于对负互为镜像映射在链环外部的JSJ分解上引起的动作的系统分析。 此外，我们提供了这种动作的充分条件，以推断负互为镜像扭结何时与强负互为镜像扭结同伦或同痕。 特别是，我们证明了每个纤维化的负互为镜像扭结都是强负互为镜像的，回答了Kim和Wu在2016年对Miyazaki扭结提出的问题。

[5] arXiv:2509.21274 [cn-pdf, pdf, other]

Title: On the arc index and Turaev genus of a link Show Chinese title

Title: 论链的弧指数和图尔亚夫亏格

Álvaro Del Valle Vílchez, Adam M. Lowrance

Comments: 33 pages, 19 figures

Subjects: Geometric Topology (math.GT)

We compute the arc index of an adequate link and establish bounds on the arc index of the closure of a positive 3-braid. We also conjecture an inequality between the crossing number, arc index, and Turaev genus of a link and show the conjecture is true for several infinite families of links including alternating links, links with Turaev genus one, adequate links, closures of positive 3-braids, torus links, and most Kanenobu knots. Show Chinese abstract

我们计算了适当链的弧指数，并建立了正3-辫闭合的弧指数的界限。 我们还提出了一个关于链的交叉数、弧指数和图尔亚夫亏格的不等式，并证明了该猜想对于多个无限链族成立，包括交替链、图尔亚夫亏格为1的链、适当链、正3-辫的闭合、环面链以及大多数Kanenobu链。

Cross submissions (showing 2 of 2 entries )

[6] arXiv:2509.20771 (cross-list from math.CO) [cn-pdf, pdf, other]

Title: Fat Shellable Spheres Show Chinese title

Title: 脂肪壳可剥球体

Joshua Hinman

Comments: 30 pages, 6 figures

Subjects: Combinatorics (math.CO) ; Geometric Topology (math.GT)

The fatness of a 4-polytope or 3-sphere is defined as $(f_1+f_2-20)/(f_0+f_3-10)$. We construct arbitrarily fat, strongly regular CW 3-spheres that are both shellable and dual shellable. These spheres have $f$-vectors $(\Theta(n),\Theta(n\alpha(n)),\Theta(n\alpha(n)),\Theta(n))$, where $\alpha$ is the inverse Ackermann function. Show Chinese abstract

4-多面体或3-球面的肥度定义为$(f_1+f_2-20)/(f_0+f_3-10)$。我们构造了任意肥度的强正则CW 3-球面，它们既是可剥的又是对偶可剥的。这些球面具有$f$-向量$(\Theta(n),\Theta(n\alpha(n)),\Theta(n\alpha(n)),\Theta(n))$，其中$\alpha$是反阿克曼函数。

[7] arXiv:2509.21298 (cross-list from math.AG) [cn-pdf, pdf, html, other]

Title: Strong cohomological integrality for symmetric stacks Show Chinese title

Title: 对称堆栈的强上同调整性

Lucien Hennecart, Tasuki Kinjo

Comments: 17 pages. Comments are welcome!

Subjects: Algebraic Geometry (math.AG) ; Geometric Topology (math.GT) ; Representation Theory (math.RT)

We prove a strong form of the cohomological integrality theorem, decomposing the cohomology of smooth symmetric stacks as the cohomological Hall induction of the intersection cohomology of the good moduli spaces of stacks of graded points. This generalizes the previous result by the second author together with Bu--Davison--Ib\'a\~nez Nu\~nez--P\u{a}durariu to non-orthogonal stacks, and confirms a conjecture of the first author that the algebraic BPS cohomology of the quotient stack of a symmetric representation matches the intersection cohomology group whenever it is nonzero. As a consequence, we obtain a version of the cohomological integrality theorem for general 0-shifted symplectic stacks with good moduli spaces, as well as for the character stacks of general compact oriented $3$-manifolds with reductive gauge groups. As an application, we prove Halpern-Leistner's conjecture on the purity of the Borel--Moore homology of $0$-shifted symplectic stacks admitting proper good moduli spaces. Our proof combines a cohomological bound for the algebraic BPS cohomology, due to the first author and based on Efimov's lemma, with a vanishing-cycle argument due to the second author in collaboration with Bu--Davison--Ib\'a\~nez Nu\~nez--P\u{a}durariu. Show Chinese abstract

我们证明了上同调整性定理的一个强形式，将光滑对称堆栈的上同调分解为分次点堆栈的好模空间的交上同调的上同调Hall诱导。 这推广了第二作者与Bu--Davison--Ibáñez Nuñez--Pădurariu之前的成果到非正交堆栈，并验证了第一作者的一个猜想，即对称表示商堆栈的代数BPS上同调在非零时与交上同调群相等。 作为结果，我们得到了具有良好模空间的一般0-偏移辛堆栈以及具有半单规范群的一般紧致定向$3$-流形的特征堆栈的上同调整性定理的一个版本。 作为应用，我们证明了Halpern-Leistner关于承认适当良好模空间的$0$-偏移辛堆栈的Borel--Moore同调纯度猜想。 我们的证明结合了第一作者提出的代数BPS上同调的上同调界，该界基于Efimov引理，以及第二作者与Bu--Davison--Ibáñez Nuñez--Pădurariu合作提出的消失循环论证。

Replacement submissions (showing 6 of 6 entries )

[8] arXiv:2012.07184 (replaced) [cn-pdf, pdf, other]

Title: Trivalent vertices and bordered knot Floer homology in the standard basis Show Chinese title

Title: 三价顶点和标准基中的有边链环同调

Andrew Manion

Comments: 118 pages; 49 figures. This version accepted for publication in Geometry & Topology

Subjects: Geometric Topology (math.GT) ; Quantum Algebra (math.QA) ; Representation Theory (math.RT)

We define new algebras, local bimodules, and bimodule maps in the spirit of Ozsvath-Szabo's bordered knot Floer homology. We equip them with the structure of 2-representations of the categorified negative half U^- of U_q(gl(1|1)), 1-morphisms of such, and 2-morphisms respectively, and show that they categorify representations of U_q(gl(1|1)^-) and maps between them. Unlike with Ozsvath-Szabo's algebras, the algebras considered here can be built from a higher tensor product operation recently introduced by Rouquier and the author. Our bimodules are all motivated by holomorphic disk counts in Heegaard diagrams; for positive and negative crossings, the bimodules can also be expressed as mapping cones involving a singular-crossing bimodule and the identity bimodule. In fact, they arise from an action of the monoidal category of Soergel bimodules via Rouquier complexes in the usual way, the first time (to the author's knowledge) such an expression has been obtained for braiding bimodules in Heegaard Floer homology. Furthermore, the singular crossing bimodule naturally factors into two bimodules for trivalent vertices; such bimodules have not appeared in previous bordered-Floer approaches to knot Floer homology. The action of the Soergel category comes from an action of categorified quantum gl(2) on the 2-representation 2-category of U^- in line with the ideas of skew Howe duality, where the trivalent vertex bimodules are associated to 1-morphisms E, F in categorified quantum gl(2). Show Chinese abstract

我们定义了新的代数、局部双模和双模映射，其精神源于Ozsvath-Szabo的边界纽结Floer同调。 我们为它们赋予了U_q(gl(1|1))的范畴化负半部分U^-的2表示结构，分别是这样的1-态射和2-态射，并证明它们对U_q(gl(1|1)^-)的表示及其之间的映射进行了范畴化。 与Ozsvath-Szabo的代数不同，这里考虑的代数可以由Rouquier和作者最近引入的更高张量积运算构建。 我们的双模都受到Heegaard图中全纯盘计数的启发；对于正交叉和负交叉，双模也可以表示为涉及奇异交叉双模和恒等双模的映射锥。 事实上，它们是通过Rouquier复形以通常方式从Soergel双模的单子范畴作用得到的，这是（据作者所知）首次在Heegaard Floer同调中为辫子双模获得这种表达。 此外，奇异交叉双模自然分解为两个三元顶点的双模；这样的双模在之前关于纽结Floer同调的边界-Floer方法中并未出现。 Soergel范畴的作用来自于范畴化量子gl(2)对U^-的2表示2-范畴的作用，这与斜Howe对偶的思想一致，其中三元顶点双模与范畴化量子gl(2)中的1-态射E, F相关联。

[9] arXiv:2501.00250 (replaced) [cn-pdf, pdf, other]

Title: On the Bonahon--Wong--Yang invariants of pseudo-Anosov maps Show Chinese title

Title: 关于伪Anosov映射的Bonahon--Wong--Yang不变量

Stavros Garoufalidis, Tao Yu

Comments: 30 pages

Subjects: Geometric Topology (math.GT) ; High Energy Physics - Theory (hep-th)

We conjecture (and prove for once-punctured torus bundles) that the Bonahon--Wong--Yang invariants of pseudo-Anosov homeomorphisms of a punctured surface at roots of unity coincide with the 1-loop invariant of their mapping torus at roots of unity. This explains the topological invariance of the BWY invariants and how their volume conjecture, to all orders, and with exponentially small terms included, follows from the quantum modularity conjecture. Using the numerical methods of Zagier and the first author, we illustrate how to efficiently compute the invariants and their asymptotics to arbitrary order in perturbation theory, using as examples the $LR$ and the $LLR$ pseudo-Anosov monodromies of the once-punctured torus. Finally, we introduce descendant versions of the 1-loop and BWY invariants and conjecture (and numerically check for pseudo-Anosov monodromies of $L/R$-length at most 5) that they are related by a Fourier transform. This edition includes statements and proofs for roots of unity of all order, even and odd. Show Chinese abstract

我们猜想（并证明对于一次穿孔环面丛）在单位根处，穿孔曲面的伪Anosov同胚的Bonahon--Wong--Yang不变量与它们的映射环面在单位根处的1圈不变量一致。 这解释了BWY不变量的拓扑不变性以及它们的体积猜想，包括所有阶次和指数小项，如何从量子模猜想中得出。 使用Zagier和第一作者的数值方法，我们说明如何高效地计算不变量及其微扰理论中任意阶的渐近行为，以$LR$和$LLR$一次穿孔环面的伪Anosov单色为例。 最后，我们引入1圈和BWY不变量的后代版本，并猜想（并通过$L/R$长度最多为5的伪Anosov单色进行数值验证）它们通过傅里叶变换相关联。 本版包含了所有阶次（偶数和奇数）单位根的陈述和证明。

[10] arXiv:2503.05151 (replaced) [cn-pdf, pdf, html, other]

Title: Revised note on surface-link of trivial components Show Chinese title

Title: 关于平凡分量的曲面链的修订注记

Akio Kawauchi

Comments: An observation on the proof of Lemma 1.2 is added

Subjects: Geometric Topology (math.GT)

It is shown that a surface-link of ribbon surface-knot components is a ribbon surface-link if and only if it is a surface-link producing a ribbon surface-link by surgery along a self-trivial 1-handle system. This corrects an earlier statement. This result makes a corrected proof for the claim that every surface-link of trivial surface-knot components with at most one aspheric component is a ribbon surface-link. For non-ribbon surface-links of trivial components with at least two aspheric components constructed in a previous note, it adds the new property that non-ribbonability continues through surgery along any self-trivial 1-handle system. Show Chinese abstract

表明由带边面纽结分量组成的面链环是一个带边面链环当且仅当通过沿自平凡1-柄系统进行手术可以生成一个带边面链环。 这更正了先前的陈述。 这一结果为以下说法提供了更正后的证明，即每个具有至多一个非球面分量的平凡面纽结分量的面链环是一个带边面链环。 对于在之前的一篇笔记中构造的具有至少两个非球面分量的平凡分量的非带边面链环，它增加了新的性质，即非带边性可以通过任何自平凡1-柄系统进行手术继续存在。

[11] arXiv:2506.09577 (replaced) [cn-pdf, pdf, html, other]

Title: On the multiplicity of Knot Floer order under cabling Show Chinese title

Title: 关于缆绳下结的Floer阶的多重性

David Suchodoll

Subjects: Geometric Topology (math.GT)

The knot Floer order $\operatorname{Ord}(K)$ is a knot invariant derived from knot Floer homology that provides bounds on many other invariants, such as the bridge index $\operatorname{br}(K)$ for which $\operatorname{Ord}(K) + 1 \leq \operatorname{br}(K)$. For all $(p,q)$-cables of L-space knots, we show that $\operatorname{Ord}(K) + 1$ is multiplicative in $p$ when $g(K) > 1$, and the same holds for $g(K) = 1$ provided $q > 2p$. We also compute the knot Floer order in the range $q < 2p$, thereby determining $\operatorname{Ord}(K_{p,q})$ in terms of $\operatorname{Ord}(K)$ for all cables of L-space knots. We establish upper bounds under cabling for $\operatorname{Ord}(K)$ and discuss potential applications to a conjecture by Krishna and Morton, proving that the braid index of an L-space cable appears as an exponent in its Alexander polynomial if it does for its companion, provided $\operatorname{Ord}(K)+1$ is multiplicative. Show Chinese abstract

结弗洛尔序$\operatorname{Ord}(K)$是一种从结弗洛尔同调中导出的结不变量，它为许多其他不变量提供了界限，例如桥指数$\operatorname{br}(K)$，其中$\operatorname{Ord}(K) + 1 \leq \operatorname{br}(K)$。 对于所有 L-space 纽结的$(p,q)$电缆，我们证明当$g(K) > 1$时，$\operatorname{Ord}(K) + 1$在$p$中是可乘的，且当$q > 2p$时，$g(K) = 1$也具有相同性质。 我们还计算了范围内的结 Floer 顺序$q < 2p$，从而根据$\operatorname{Ord}(K)$确定了 L-空间结的所有缆绳的$\operatorname{Ord}(K_{p,q})$。 我们在缆绳下建立了$\operatorname{Ord}(K)$的上界，并讨论了对 Krishna 和 Morton 猜想的潜在应用，证明如果 L-空间缆绳的辫子指数在其亚历山大多项式中作为指数出现，则其伴随者也是如此，前提是$\operatorname{Ord}(K)+1$是可乘的。

[12] arXiv:2408.12520 (replaced) [cn-pdf, pdf, html, other]

Title: Center of stated $\mathrm{SL}(n)$-skein algebras Show Chinese title

Title: 陈述的$\mathrm{SL}(n)$-链环中心

Hiroaki Karuo, Zhihao Wang

Comments: 72 pages, 12 figures; added Appendix B, to appear in Trans. Amer. Math. Soc

Subjects: Quantum Algebra (math.QA) ; Geometric Topology (math.GT) ; Representation Theory (math.RT)

In the paper, we show some properties of (reduced) stated $\mathrm{SL}(n)$-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated $\mathrm{SL}(n)$-skein algebras into quantum tori appearing in higher Teichm\"uller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can understand the representation theory of (reduced) stated $\mathrm{SL}(n)$-skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access to the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially. Show Chinese abstract

在本文中，我们展示了与本质上边界为pb的曲面相关的（约化）带标记的$\mathrm{SL}(n)$-辫代数的一些性质，特别是它们的中心、在中心上的有限生成性以及它们的PI次数。证明基于量子迹映射，以及（约化）带标记的$\mathrm{SL}(n)$-辫代数嵌入到高Teichmüller理论中出现的量子环面。得益于[BG02, FKBL19]中的唯一性定理，我们可以理解（约化）带标记的$\mathrm{SL}(n)$-辫代数的表示理论。此外，应用范围超出了低维拓扑。例如，我们可以接触到无限制量子模空间代数的表示理论，以及潜在的量子高簇代数的表示理论。

[13] arXiv:2504.16962 (replaced) [cn-pdf, pdf, html, other]

Title: Morse-Bott-Smale chain complex Show Chinese title

Title: 莫尔斯-博特-斯梅尔链复形

Ryuma Orita, Kanon Yashiro

Comments: 30 pages, 9 figures

Subjects: Algebraic Topology (math.AT) ; Dynamical Systems (math.DS) ; Geometric Topology (math.GT)

Banyaga and Hurtubise defined the Morse-Bott-Smale chain complex as a quotient of a large chain complex by introducing five degeneracy relations. However, their five degeneracy relations are in fact redundant. In the present paper, we unify these five conditions into a single degeneracy condition and resolve the issue of the well-definedness of the Morse-Bott-Smale chain complex. This provides an appropriate definition of the Morse-Bott homology and more computable examples. Moreover, we show that our chain complex for a Morse-Smale function is quasi-isomorphic to the usual Morse-Smale-Witten chain complex. As a consequence, we obtain an alternative proof of the Morse Homology Theorem. Show Chinese abstract

Banyaga和Hurtubise将Morse-Bott-Smale链复形定义为通过引入五个退化关系对一个大链复形的商。 然而，他们的五个退化关系实际上是冗余的。 在本文中，我们将这五个条件统一为一个退化条件，并解决了Morse-Bott-Smale链复形的良定义性问题。 这提供了Morse-Bott同调的一个适当定义以及更多可计算的例子。 此外，我们证明了对于一个Morse-Smale函数，我们的链复形与通常的Morse-Smale-Witten链复形是拟同构的。 作为结果，我们得到了Morse同调定理的一个替代证明。