When working with (multi-parameter) persistence modules, one usually makes some type of tameness assumption in order to obtain better control over their algebraic behavior. One such notion is Ezra Millers notion of finite encodability, which roughly states that a persistence module can be obtained by pulling back a finite dimensional persistence module over a finite poset. From the perspective of homological algebra, finitely encodable persistence have an inconvenient property: They do not form an abelian category. Here, we prove that if one restricts to such persistence modules which can be constructed in terms of topologically closed and sufficiently constructible (piecewise linear, semi-algebraic, etc.) upsets then abelianity can be restored.
{"title":"Notes on abelianity of categories of finitely encoded persistence modules","authors":"Lukas Waas","doi":"arxiv-2407.08666","DOIUrl":"https://doi.org/arxiv-2407.08666","url":null,"abstract":"When working with (multi-parameter) persistence modules, one usually makes\u0000some type of tameness assumption in order to obtain better control over their\u0000algebraic behavior. One such notion is Ezra Millers notion of finite\u0000encodability, which roughly states that a persistence module can be obtained by\u0000pulling back a finite dimensional persistence module over a finite poset. From\u0000the perspective of homological algebra, finitely encodable persistence have an\u0000inconvenient property: They do not form an abelian category. Here, we prove\u0000that if one restricts to such persistence modules which can be constructed in\u0000terms of topologically closed and sufficiently constructible (piecewise linear,\u0000semi-algebraic, etc.) upsets then abelianity can be restored.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ilya Trofimov, Daria Voronkova, Eduard Tulchinskii, Evgeny Burnaev, Serguei Barannikov
We propose a new topological tool for computer vision - Scalar Function Topology Divergence (SFTD), which measures the dissimilarity of multi-scale topology between sublevel sets of two functions having a common domain. Functions can be defined on an undirected graph or Euclidean space of any dimensionality. Most of the existing methods for comparing topology are based on Wasserstein distance between persistence barcodes and they don't take into account the localization of topological features. On the other hand, the minimization of SFTD ensures that the corresponding topological features of scalar functions are located in the same places. The proposed tool provides useful visualizations depicting areas where functions have topological dissimilarities. We provide applications of the proposed method to 3D computer vision. In particular, experiments demonstrate that SFTD improves the reconstruction of cellular 3D shapes from 2D fluorescence microscopy images, and helps to identify topological errors in 3D segmentation.
{"title":"Scalar Function Topology Divergence: Comparing Topology of 3D Objects","authors":"Ilya Trofimov, Daria Voronkova, Eduard Tulchinskii, Evgeny Burnaev, Serguei Barannikov","doi":"arxiv-2407.08364","DOIUrl":"https://doi.org/arxiv-2407.08364","url":null,"abstract":"We propose a new topological tool for computer vision - Scalar Function\u0000Topology Divergence (SFTD), which measures the dissimilarity of multi-scale\u0000topology between sublevel sets of two functions having a common domain.\u0000Functions can be defined on an undirected graph or Euclidean space of any\u0000dimensionality. Most of the existing methods for comparing topology are based\u0000on Wasserstein distance between persistence barcodes and they don't take into\u0000account the localization of topological features. On the other hand, the\u0000minimization of SFTD ensures that the corresponding topological features of\u0000scalar functions are located in the same places. The proposed tool provides\u0000useful visualizations depicting areas where functions have topological\u0000dissimilarities. We provide applications of the proposed method to 3D computer\u0000vision. In particular, experiments demonstrate that SFTD improves the\u0000reconstruction of cellular 3D shapes from 2D fluorescence microscopy images,\u0000and helps to identify topological errors in 3D segmentation.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A simply connected topological space is called emph{rationally elliptic} if the rank of its total homotopy group and its total (co)homology group are both finite. A well-known Hilali conjecture claims that for a rationally elliptic space its homotopy rank emph{does not exceed} its (co)homology rank. In this paper, after recalling some well-known fundamental properties of a rationally elliptic space and giving some important examples of rationally elliptic spaces and rationally elliptic singular complex algebraic varieties for which the Hilali conjecture holds, we give some revised formulas and some conjectures. We also discuss some topics such as mixd Hodge polynomials defined via mixed Hodge structures on cohomology group and the dual of the homotopy group, related to the ``Hilali conjecture emph{modulo product}", which is an inequality between the usual homological Poincar'e polynomial and the homotopical Poincar'e polynomial.
{"title":"Hilali conjecture and complex algebraic varieties","authors":"Shoji Yokura","doi":"arxiv-2407.06548","DOIUrl":"https://doi.org/arxiv-2407.06548","url":null,"abstract":"A simply connected topological space is called emph{rationally elliptic} if\u0000the rank of its total homotopy group and its total (co)homology group are both\u0000finite. A well-known Hilali conjecture claims that for a rationally elliptic\u0000space its homotopy rank emph{does not exceed} its (co)homology rank. In this\u0000paper, after recalling some well-known fundamental properties of a rationally\u0000elliptic space and giving some important examples of rationally elliptic spaces\u0000and rationally elliptic singular complex algebraic varieties for which the\u0000Hilali conjecture holds, we give some revised formulas and some conjectures. We\u0000also discuss some topics such as mixd Hodge polynomials defined via mixed Hodge\u0000structures on cohomology group and the dual of the homotopy group, related to\u0000the ``Hilali conjecture emph{modulo product}\", which is an inequality between\u0000the usual homological Poincar'e polynomial and the homotopical Poincar'e\u0000polynomial.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141587089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the $infty$-category of global spaces is equivalent to the homotopy localization of the $infty$-category of sheaves on the site of separated differentiable stacks, following a philosophy proposed by Gepner-Henriques. We further prove that this $infty$-category of sheaves is a cohesive $infty$-topos and that it fully faithfully contains the singular-cohesive $infty$-topos of Sati-Schreiber.
{"title":"Global spaces and the homotopy theory of stacks","authors":"Adrian Clough, Bastiaan Cnossen, Sil Linskens","doi":"arxiv-2407.06877","DOIUrl":"https://doi.org/arxiv-2407.06877","url":null,"abstract":"We show that the $infty$-category of global spaces is equivalent to the\u0000homotopy localization of the $infty$-category of sheaves on the site of\u0000separated differentiable stacks, following a philosophy proposed by\u0000Gepner-Henriques. We further prove that this $infty$-category of sheaves is a\u0000cohesive $infty$-topos and that it fully faithfully contains the\u0000singular-cohesive $infty$-topos of Sati-Schreiber.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141587094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Chang-Skjelbred method computes the cohomology of a suitable space with a torus action from its equivariant one-skeleton. We show that, under certain restrictions on the cohomological torsion, the integral cohomology is encoded in the one-skeleton even in the presence of arbitrary disconnected isotropy groups. We provide applications to Hamiltonian actions as well as to the GKM case. In the latter, our results lead to a modification of the GKM formula for graph cohomology.
{"title":"On integral Chang-Skjelbred computations with disconnected isotropy groups","authors":"Leopold Zoller","doi":"arxiv-2407.03052","DOIUrl":"https://doi.org/arxiv-2407.03052","url":null,"abstract":"The Chang-Skjelbred method computes the cohomology of a suitable space with a\u0000torus action from its equivariant one-skeleton. We show that, under certain\u0000restrictions on the cohomological torsion, the integral cohomology is encoded\u0000in the one-skeleton even in the presence of arbitrary disconnected isotropy\u0000groups. We provide applications to Hamiltonian actions as well as to the GKM\u0000case. In the latter, our results lead to a modification of the GKM formula for\u0000graph cohomology.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the unit of the Quillen pair ${mathfrak{L}}colon {bf sset}rightleftarrows {bf cdgl}colon {langle,cdot,rangle}$ given by the model and realization functor is, up to homotopy, the Bousfield-Kan ${mathbb{Q}}$-completion.
{"title":"A Lie characterization of the Bousfield-Kan ${mathbb{Q}}$-completion and ${mathbb{Q}}$-good spaces","authors":"Yves Félix, Mario Fuentes, Aniceto Murillo","doi":"arxiv-2407.02812","DOIUrl":"https://doi.org/arxiv-2407.02812","url":null,"abstract":"We prove that the unit of the Quillen pair ${mathfrak{L}}colon {bf\u0000sset}rightleftarrows {bf cdgl}colon {langle,cdot,rangle}$ given by the\u0000model and realization functor is, up to homotopy, the Bousfield-Kan\u0000${mathbb{Q}}$-completion.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141547069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to study convergence of Bousfield-Kan completions with respect to the 1-excisive approximation of the identity functor and exotic convergence of the Taylor tower of the identity functor, for algebras over operads in spectra centered away from the null object. In Goodwillie's homotopy functor calculus, being centered away from the null object amounts to doing homotopy theory and functor calculus in the retractive setting.
{"title":"Functor calculus completions for retractive operadic algebras in spectra","authors":"Matthew B. Carr, John E. Harper","doi":"arxiv-2407.01819","DOIUrl":"https://doi.org/arxiv-2407.01819","url":null,"abstract":"The aim of this paper is to study convergence of Bousfield-Kan completions\u0000with respect to the 1-excisive approximation of the identity functor and exotic\u0000convergence of the Taylor tower of the identity functor, for algebras over\u0000operads in spectra centered away from the null object. In Goodwillie's homotopy\u0000functor calculus, being centered away from the null object amounts to doing\u0000homotopy theory and functor calculus in the retractive setting.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christian Carrick, Jack Morgan Davies, Sven van Nigtevecht
The synthetic analogue functor $nu$ from spectra to synthetic spectra does not preserve all limits. In this paper, we give a necessary and sufficient criterion for $nu$ to preserve the global sections of a derived stack. Even when these conditions are not satisfied, our framework still yields synthetic spectra that implement the descent spectral sequence for the structure sheaf, thus placing descent spectral sequences on good footing in the $infty$-category of synthetic spectra. As an example, we introduce a new $mathrm{MU}$-synthetic spectrum $mathrm{Smf}$.
{"title":"Descent spectral sequences through synthetic spectra","authors":"Christian Carrick, Jack Morgan Davies, Sven van Nigtevecht","doi":"arxiv-2407.01507","DOIUrl":"https://doi.org/arxiv-2407.01507","url":null,"abstract":"The synthetic analogue functor $nu$ from spectra to synthetic spectra does\u0000not preserve all limits. In this paper, we give a necessary and sufficient\u0000criterion for $nu$ to preserve the global sections of a derived stack. Even\u0000when these conditions are not satisfied, our framework still yields synthetic\u0000spectra that implement the descent spectral sequence for the structure sheaf,\u0000thus placing descent spectral sequences on good footing in the\u0000$infty$-category of synthetic spectra. As an example, we introduce a new\u0000$mathrm{MU}$-synthetic spectrum $mathrm{Smf}$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the loop and suspension functors on the category of augmented $mathbb{E}_n$-algebras. One application is to the formality of the cochain algebra of the $n$-sphere. We show that it is formal as an $mathbb{E}_n$-algebra, also with coefficients in general commutative ring spectra, but rarely $mathbb{E}_{n+1}$-formal unless the coefficients are rational. Along the way we show that the free functor from operads in spectra to monads in spectra is fully faithful on a nice subcategory of operads which in particular contains the stable $mathbb{E}_n$-operads for finite $n$. We use this to interpret our results on loop and suspension functors of augmented algebras in operadic terms.
{"title":"Formality of $mathbb{E}_n$-algebras and cochains on spheres","authors":"Gijs Heuts, Markus Land","doi":"arxiv-2407.00790","DOIUrl":"https://doi.org/arxiv-2407.00790","url":null,"abstract":"We study the loop and suspension functors on the category of augmented\u0000$mathbb{E}_n$-algebras. One application is to the formality of the cochain\u0000algebra of the $n$-sphere. We show that it is formal as an\u0000$mathbb{E}_n$-algebra, also with coefficients in general commutative ring\u0000spectra, but rarely $mathbb{E}_{n+1}$-formal unless the coefficients are\u0000rational. Along the way we show that the free functor from operads in spectra\u0000to monads in spectra is fully faithful on a nice subcategory of operads which\u0000in particular contains the stable $mathbb{E}_n$-operads for finite $n$. We use\u0000this to interpret our results on loop and suspension functors of augmented\u0000algebras in operadic terms.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In topological data analysis (TDA), a longstanding challenge is to recognize underlying geometric structures in noisy data. One motivating examples is the shape of a point cloud in Euclidean space given by image. Carlsson et al. proposed a method to detect topological features in point clouds by first filtering by density and then applying persistent homology. Later more refined methods have been developed, such as the degree Rips complex of Lesnick and Wright and the multicover bifiltration. In this paper we introduce the dual Degree Cech bifiltration, a Prohorov stable bicomplex of a point cloud in a metric space with the point cloud itself as vertex set. It is of the same homotopy type as the Measure Dowker bifiltration of Hellmer and Spali'nski but it has a different vertex set. The dual Degree Cech bifiltration can be constructed both in an ambient and an intrinsic way. The intrinsic dual Degree Cech bifiltration is a $(1,2)$-intereaved with the ambent dual Degree Cech bifiltration in the distance parameter. This interleaving can be used to leverage a stability result for the intrinsically defined dual Degree Cech bifiltration. This stability result recently occured in work by Hellmer and Spali'nski.
在拓扑数据分析（TDA）中，一个长期存在的挑战是识别噪声数据中潜在的几何结构。其中一个激励性的例子是图像给出的欧几里得空间中点云的形状。Carlsson 等人提出了一种检测点云拓扑特征的方法，首先通过密度过滤，然后应用持久同源性。后来，人们又开发出了更精细的方法，如莱斯尼克和赖特的度里普斯复合法以及多覆盖分层法。在本文中，我们介绍了对偶度 Cech 双分层，即以点云本身为顶点集的对称空间中点云的普罗霍罗夫稳定双复数。它与赫尔默和斯帕利斯基的度量道克二分层属于同一同调类型，但它的顶点集不同。对偶 Degree Cech 双分层可以通过环境和内在两种方式构造。内在的对偶 Degree Cech 双分层与外在的对偶 Degree Cech 双分层在距离参数上是$(1,2)$交错的。这种交错可以用来利用内在定义的双度切赫分层的稳定性结果。这一稳定性结果最近出现在 Hellmer 和 Spali'nski 的研究中。
{"title":"The Dual Degree Cech Bifiltration","authors":"Morten Brun","doi":"arxiv-2407.00477","DOIUrl":"https://doi.org/arxiv-2407.00477","url":null,"abstract":"In topological data analysis (TDA), a longstanding challenge is to recognize\u0000underlying geometric structures in noisy data. One motivating examples is the\u0000shape of a point cloud in Euclidean space given by image. Carlsson et al.\u0000proposed a method to detect topological features in point clouds by first\u0000filtering by density and then applying persistent homology. Later more refined\u0000methods have been developed, such as the degree Rips complex of Lesnick and\u0000Wright and the multicover bifiltration. In this paper we introduce the dual\u0000Degree Cech bifiltration, a Prohorov stable bicomplex of a point cloud in a\u0000metric space with the point cloud itself as vertex set. It is of the same\u0000homotopy type as the Measure Dowker bifiltration of Hellmer and Spali'nski but\u0000it has a different vertex set. The dual Degree Cech bifiltration can be constructed both in an ambient and\u0000an intrinsic way. The intrinsic dual Degree Cech bifiltration is a\u0000$(1,2)$-intereaved with the ambent dual Degree Cech bifiltration in the\u0000distance parameter. This interleaving can be used to leverage a stability\u0000result for the intrinsically defined dual Degree Cech bifiltration. This\u0000stability result recently occured in work by Hellmer and Spali'nski.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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