代数拓扑

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新提交 (展示 2 之 2 条目 )

[1] arXiv:2508.02914 [中文pdf, pdf, html, 其他]

标题： 2-范畴论在多参数持久性中的基础作用 显示英文标题

标题： 2-Categorical Foundations for Multiparameter Persistence

Mauricio Angel

评论： 34页

主题： 代数拓扑 (math.AT) ; 微分几何 (math.DG)

本文介绍了一种使用2范畴结构的多参数持久性的新方法。 我们开发了一个框架，能够捕捉过滤参数之间的层次交互，克服了传统持久性模块的基本局限性。 我们的2范畴模型产生了新的不变量，这些不变量能够有效表征多维拓扑特征，同时保持计算可行性。 我们证明了这些不变量的稳定性定理，并通过在基因组学和复杂网络分析中的应用展示了它们的有效性。 显示英文摘要

This paper introduces a novel approach to multi-parameter persistence using 2-categorical structures. We develop a framework that captures hierarchical interactions between filter parameters, overcoming fundamental limitations of traditional persistence modules. Our 2-categorical model yields new invariants that effectively characterize multidimensional topological features while maintaining computational tractability. We prove stability theorems for these invariants and demonstrate their effectiveness through applications in genomics and complex network analysis.

[2] arXiv:2508.03224 [中文pdf, pdf, html, 其他]

标题： 交叠同伦，精化和粗化 显示英文标题

标题： Intersection homotopy, refinements and coarsenings

Martintxo Saralegi-Aranguren, Daniel Tanré

主题： 代数拓扑 (math.AT) ; 几何拓扑 (math.GT)

在以前的工作中，我们研究了与Goresky和MacPherson的perversity以及过滤空间相关的交集同伦群。 它们被定义为P. Gajer引入的单纯集的同伦群。 我们将其具体化到Siebenmann的局部锥形空间（称为CS集），并在常规部分保持不变时建立了它们的拓扑不变性。 在这里，我们考虑由两个CS集结构组成的粗化，同一拓扑空间上的分层结构，其中一个的分层是另一个分层的并集。 我们为其赋予一个一般的perversity及其前推，其中“一般”一词意味着perversity是在分层的偏序集上定义的，而不仅仅是根据其余维数。 如果perversity满足类似于Goresky和MacPherson原始perversity的生长性质，我们在上述对常规部分的限制下，也找到了粗化交集同伦群的不变性定理。 在某些情况下，当奇异分层在粗化中变为常规分层时，也展示了Thom-Mather空间的不变性。 显示英文摘要

In previous works, we studied intersection homotopy groups associated to a Goresky and MacPherson perversity and a filtered space. They are defined as the homotopy groups of simplicial sets introduced by P. Gajer. We particularized to locally conical spaces of Siebenmann (called CS sets) and established a topological invariance for them when the regular part remains unchanged. Here, we consider coarsenings, made of two structures of CS sets on the same topological space, the strata of one being a union of strata of the other. We endow them with a general perversity and its pushforward, where the adjective ``general'' means that the perversities are defined on the poset of the strata and not only according to their codimension. If the perversity verifies a growing property analogous to that of the original perversities of Goresky and MacPherson, we also find an invariance theorem for the intersection homotopy groups of a coarsening, under the above restriction on the regular parts. An invariance is shown too in some cases where singular strata become regular in the coarsening, for Thom-Mather spaces.

交叉提交 (展示 1 之 1 条目 )

[3] arXiv:2508.03621 (交叉列表自 math.KT) [中文pdf, pdf, html, 其他]

标题： 一个真实的$G$-谱对于剪切和粘贴$K$-理论的$G$-流形 显示英文标题

标题： A genuine $G$-spectrum for the cut-and-paste $K$-theory of $G$-manifolds

Maxine Calle, David Chan

评论： 15页，欢迎提出意见！

主题： K理论与同调 (math.KT) ; 代数拓扑 (math.AT)

最近的工作已将剪切相容性$K$-理论应用于研究流形的经典剪切和粘贴($SK$)不变量。 本文证明了这样一个猜想，即等变$SK$-流形的平方$K$-理论作为真正的$G$-谱的空间固定点出现。 我们的方法利用了谱Mackey函子的框架，作为真正$G$-谱的模型，我们主要的技术结果是使用平方$K$-理论构建谱Mackey函子的一般过程。 显示英文摘要

Recent work has applied scissors congruence $K$-theory to study classical cut-and-paste ($SK$) invariants of manifolds. This paper proves the conjecture that the squares $K$-theory of equivariant $SK$-manifolds arises as the fixed points of a genuine $G$-spectrum. Our method utilizes the framework of spectral Mackey functors as models for genuine $G$-spectra, and our main technical result is a general procedure for constructing spectral Mackey functors using squares $K$-theory.

替换提交 (展示 8 之 8 条目 )

[4] arXiv:1803.11014 (替换) [中文pdf, pdf, html, 其他]

标题： p局部稳定Adams猜想：勘误 显示英文标题

标题： The p-local stable Adams conjecture: Erratum

Prasit Bhattacharya, Nitu Kitchloo

主题： 代数拓扑 (math.AT)

这份更正修正了文献中与稳定 Adams 猜想相关的错误。 作为上述更正的一部分，我们还识别并修正了我们近期关于该主题的文章第 4 节中的两个错误。 我们感谢 E. Fridelander 指出这些疏漏并提出了有帮助的建议。 本更正自成一体，并包括一个附录，证明了 Friedlander 对分节纤维空间的分类结果的一个版本，如附录中所指示的对原始陈述进行了适当修改。 显示英文摘要

This erratum remedies errors in the literature pertaining to the stable Adams conjecture. As part of the above corrections, we also identify and fix two errors in section 4 of our recent article on the subject. We thank E. Fridelander for flagging these oversights and for offering helpful suggestions. This erratum is self-contained and also includes an appendix proving a version of Friedlander's classification result for sectioned fibrations of Gamma-spaces, after making appropriate changes to the original statement as indicated in the appendix.

[5] arXiv:2310.17494 (替换) [中文pdf, pdf, html, 其他]

标题： 时间序列数据的拓扑特征选择 显示英文标题

标题： Topological feature selection for time series data

Peter Bubenik, Johnathan Bush

评论： 27页。为均值梯度路径添加了统计保证（大数定律，中心极限定理）；扩展了应用部分；进行了各种小的编辑和更正

主题： 代数拓扑 (math.AT) ; 优化与控制 (math.OC)

我们使用应用拓扑学的工具对时间序列数据进行特征选择。 我们开发了一种方法，用于对多变量时间序列中的变量进行评分，该方法反映了它们对相应点云的拓扑特征的贡献。 我们的方法在标准几何单纯形中产生一条分段线性Lipschitz梯度路径，从质心开始，此时变量权重相等，最终到达得分。 向输入数据添加高斯扰动并取期望值，得到的均值梯度路径满足大数定律和中心极限定理。 我们的理论源于对线虫C. elegans神经元活动的分析，我们的方法选择了一组有信息量的神经元，以优化协调动力学。 显示英文摘要

We use tools from applied topology for feature selection on time series data. We develop a method for scoring the variables in a multivariate time series that reflects their contributions to the topological features of the corresponding point cloud. Our approach produces a piecewise-linear Lipschitz gradient path in the standard geometric simplex that starts at the barycenter, which weights the variables equally, and ends at the score. Adding Gaussian perturbations to the input data and taking expectations results in a mean gradient path that satisfies a strong law of large numbers and central limit theorem. Our theory is motivated by the analysis of the neuronal activities of the nematode C. elegans, and our method selects an informative subset of the neurons that optimizes the coordinated dynamics.

[6] arXiv:2311.16881 (替换) [中文pdf, pdf, html, 其他]

标题： (非-)消没结果关于自由群上简单外函子之间的扩张 显示英文标题

标题： (Non-)vanishing results for extensions between simple outer functors on free groups

Louis Hainaut

评论： 21页，1表，2图。出版前的最终版本，主要为风格修改。有关相关算法和计算，请参见https://github.com/louishainaut/Ext-Outer-Functors

主题： 代数拓扑 (math.AT) ; 范畴论 (math.CT)

在本文中，我们研究了自由群上多项式外函子范畴的上同调性质，这些函子是从有限生成自由群的范畴到有理向量空间的范畴的函子，它们将所有内自同构映射为恒等态射，并满足某种多项式性条件。 更准确地说，我们证明了简单多项式外函子之间的Ext群的消失和非消失结果。 这项工作受到Vespa先前关于从有限生成自由群到有理向量空间的所有多项式函子范畴的结果的启发；特别是，从她的结果可以得出，在这个更大的范畴中，简单函子之间的Ext群总是集中在特定的单一次数上。 我们的主要结果表明，当我们转向多项式外函子的全子范畴时，简单函子之间的Ext群在该特定次数之外有时是非平凡的。 显示英文摘要

In this article we study cohomological properties of the category of polynomial outer functors on free groups, which are the functors from the category of finitely generated free groups to the category of rational vector spaces which send all inner automorphisms to the identity morphism, and which satisfy a certain polynomiality property. More precisely, we prove vanishing and non-vanishing results for the Ext groups between simple polynomial outer functors. This work is inspired by an earlier result of Vespa for the category of all polynomial functors from finitely generated free groups to rational vector spaces; it follows in particular from her results that, in this larger category, the Ext groups between simple functors are always concentrated in a specific single degree. Our main results show that, when we pass to the full subcategory of polynomial outer functors, Ext groups between simple functors are sometimes non-trivial outside of this specific degree.

[7] arXiv:2402.12339 (替换) [中文pdf, pdf, html, 其他]

标题： 推移空间 显示英文标题

标题： Path spaces of pushouts

David Wärn

主题： 代数拓扑 (math.AT) ; 范畴论 (math.CT)

给定一个空间的范围，可以形成同伦上积，然后取由此得到的余范围的同伦下积。 我们给出这个下积的具体描述，作为一系列近似值的余极限，使用我们称之为锯齿构造的方法。 我们还得到了同伦上积的环路空间的描述。 利用锯齿构造，我们重现了Blakers-Massey定理的推广以及Bass-Serre理论中的基本结果。 我们还描述了并接的环路空间，并证明它在悬架后可以分解。 我们的构造可以在一大类$(\infty,1)$-范畴和同伦类型理论中进行解释，在那里它解决了长期以来的开放问题，即证明0型的上积是1截断的。 锯齿构造与James构造密切相关，但具有更大的普遍性。 显示英文摘要

Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the zigzag construction. We also obtain a description of loop spaces of homotopy pushouts. Using the zigzag construction, we reproduce generalisations of the Blakers-Massey theorem and fundamental results from Bass-Serre theory. We also describe the loop space of a wedge and show that it splits after suspension. Our construction can be interpreted in a large class of $(\infty,1)$-categories and in homotopy type theory, where it resolves the long-standing open problem of showing that a pushout of 0-types is 1-truncated. The zigzag construction is closely related to the James construction, but works in greater generality.

[8] arXiv:2508.01347 (替换) [中文pdf, pdf, 其他]

标题： 廉价嵌入原理：同调增长的动力学上界 显示英文标题

标题： The cheap embedding principle: Dynamical upper bounds for homology growth

Kevin Li, Clara Loeh, Marco Moraschini, Roman Sauer, Matthias Uschold

评论： 104页，标题已更正

主题： 代数拓扑 (math.AT) ; 动力系统 (math.DS) ; 群论 (math.GR) ; 算子代数 (math.OA)

我们提供了群的对数扭同调增长和贝蒂数增长的上界，用测度群理论的语言来表述。 显示英文摘要

We provide upper bounds for logarithmic torsion homology growth and Betti number growth of groups, phrased in the language of measured group theory.

[9] arXiv:2405.17113 (替换) [中文pdf, pdf, html, 其他]

标题： 有理同伦与主$G$-丛模堆的霍奇理论 显示英文标题

标题： Rational Homotopy and Hodge Theory of Moduli Stacks of principal $G$-bundles

Pedro L. del Angel R., Frank Neumann

评论： 14页

期刊参考： 新的数学分析工具及其应用——第14届ISAAC大会论文集，巴西里贝朗普雷托，2023年，{\em 数学趋势}，{\em 研究视角}，Springer--Birkh\"auser 2025年，35--49

主题： 代数几何 (math.AG) ; 代数拓扑 (math.AT)

对于半单复代数群$G$，我们使用底层拓扑堆栈的同伦理论，确定了在连通光滑复射影簇$X$上主$G$-丛的模堆栈${\mathscr B}un_{G,X}$的有理上同调和 Hodge-Tate 结构。 显示英文摘要

For a semisimple complex algebraic group $G$ we determine the rational cohomology and the Hodge-Tate structure of the moduli stack ${\mathscr B}un_{G,X}$ of principal $G$-bundles over a connected smooth complex projective variety $X$ of special type using the homotopy theory of the underlying topological stack.

[10] arXiv:2411.03257 (替换) [中文pdf, pdf, html, 其他]

标题： 谱弗洛尔理论和切向结构 显示英文标题

标题： Spectral Floer theory and tangential structures

Noah Porcelli, Ivan Smith

评论： 欢迎提出意见！v3：接受的版本，包含了审稿人的意见和建议

主题： 辛几何 (math.SG) ; 代数拓扑 (math.AT) ; K理论与同调 (math.KT)

在\cite{PS}中，对于一个稳定框架的黎曼流形$X$，我们定义了一个球面谱上的 Donaldson-Fukaya 范畴$\mathcal{F}(X;\mathbb{S})$，并发展了一种将拟同构从$\mathcal{F}(X;\mathbb{Z})$提升到$\mathcal{F}(X;\mathbb{S})$的障碍理论。 这里，我们为任何“分级切向对”$\Theta \to \Phi$定义了一个谱Donaldson-Fukaya范畴，这些空间位于$BO \to BU$上，其对象是切丛的分类映射提升到$\Theta \to \Phi$的拉格朗日子流形$L\to X$。 之前的案例对应于$\Theta = \Phi = \{\mathrm{pt}\}$。 我们将我们的障碍理论扩展到这种情形。 调整$\Theta$和$\Phi$的选择的灵活性增加了可以消除障碍的情况范围，这在相应的 bordism 理论中对 Lagrangian 嵌入的 bordism 类有应用$\Omega^{(\Theta,\Phi),\circ}_*$。我们包含了一个关于在环谱$R$上（精确）谱 Floer 理论何时存在的自含讨论，这可能具有独立的兴趣。 显示英文摘要

In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $\Theta \to \Phi$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $\Theta \to \Phi$. The previous case corresponded to $\Theta = \Phi = \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $\Theta$ and $\Phi$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory $\Omega^{(\Theta,\Phi),\circ}_*$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest.

[11] arXiv:2507.05845 (替换) [中文pdf, pdf, html, 其他]

标题： 从有理顶点算子代数的共形块中得到的模 functor 显示英文标题

标题： Modular functors from conformal blocks of rational vertex operator algebras

Chiara Damiolini, Lukas Woike

评论： 31页；v2：小幅度修改

主题： 量子代数 (math.QA) ; 数学物理 (math-ph) ; 代数几何 (math.AG) ; 代数拓扑 (math.AT)

对于顶点算子代数$V$，可以自然地按照 Frenkel-Ben-Zvi 的构造并由 Damiolini-Gibney-Tarasca 推广来定义共形块的空间。如果$V$是强有理的，这些共形块的空间在适合的代数曲线模空间上形成向量丛。在本文中，在相同的假设下，我们建立了广泛预期的拓扑结果，即共形块的空间产生一个模 functor，即在曲面操作符的扩展上的模代数。这意味着，可接受的$V$-模的范畴$\mathcal{C}_V$从零亏格曲面的拓扑中继承了一个带状 Grothendieck-Verdier 结构，甚至导致了一个模融合范畴的结构，其结构直接来自$V$的共形块的空间。作为直接结果，我们证明了来自共形块的模 functor 可以扩展为一个三维拓扑场理论，并且可以用分解同调来描述。 显示英文摘要

For a vertex operator algebra $V$, one may naturally define spaces of conformal blocks following a construction of Frenkel-Ben-Zvi generalized by Damiolini-Gibney-Tarasca. If $V$ is strongly rational, these spaces of conformal blocks form vector bundles over a suitable moduli space of algebraic curves. In this article, we establish, under the same assumptions, the widely expected topological result that the spaces of conformal blocks produce a modular functor, i.e. a modular algebra over an extension of the surface operad. This entails that the category $\mathcal{C}_V$ of admissible $V$-modules inherits from the topology of genus zero surfaces a ribbon Grothendieck-Verdier structure that leads even to the structure of a modular fusion category whose structure comes directly from the spaces of conformal blocks of $V$. As a direct consequence, we prove that the modular functor from conformal blocks extends to a three-dimensional topological field theory and comes with a description in terms of factorization homology.