Pub Date : 2023-11-10DOI: 10.1142/s1793525323500437
Maria Simkova
This paper proposes an algorithm that decides if two simply connected spaces represented by finite simplicial sets of finite $k$-type and finite dimension $d$ are homotopy equivalent. If the spaces are homotopy equivalent, the algorithm finds a homotopy equivalence between their Postnikov stages in dimension $d$. As a consequence, we get an algorithm deciding if two spaces represented by finite simplicial sets are stably homotopy equivalent.
{"title":"Are two <i>H</i>-Spaces Homotopy Equivalent? An Algorithmic View Point","authors":"Maria Simkova","doi":"10.1142/s1793525323500437","DOIUrl":"https://doi.org/10.1142/s1793525323500437","url":null,"abstract":"This paper proposes an algorithm that decides if two simply connected spaces represented by finite simplicial sets of finite $k$-type and finite dimension $d$ are homotopy equivalent. If the spaces are homotopy equivalent, the algorithm finds a homotopy equivalence between their Postnikov stages in dimension $d$. As a consequence, we get an algorithm deciding if two spaces represented by finite simplicial sets are stably homotopy equivalent.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":" July","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1142/s1793525323500425
Bruno Staffa
Stationary geodesic networks are the analogs of closed geodesics whose domain is a graph instead of a circle. We prove that for a generic Riemannian metric on a smooth manifold $M$ regular stationary geodesic nets are non-degenerate.
{"title":"Bumpy Metrics Theorem for Geodesic Nets","authors":"Bruno Staffa","doi":"10.1142/s1793525323500425","DOIUrl":"https://doi.org/10.1142/s1793525323500425","url":null,"abstract":"Stationary geodesic networks are the analogs of closed geodesics whose domain is a graph instead of a circle. We prove that for a generic Riemannian metric on a smooth manifold $M$ regular stationary geodesic nets are non-degenerate.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"84 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1142/s1793525323500371
Michael Gene Dobbins
We study the evolution of a Jordan curve on the 2-sphere by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously on the initial curve in Frechet distance in the case where the curve bisects the sphere. This even holds in the limit as time goes to infinity. This builds on Joseph Lauer's work on existence and uniqueness of solutions to the curvature flow problem on the sphere when the initial curve is not smooth.
{"title":"Continuous dependence of curvature flow on initial conditions","authors":"Michael Gene Dobbins","doi":"10.1142/s1793525323500371","DOIUrl":"https://doi.org/10.1142/s1793525323500371","url":null,"abstract":"We study the evolution of a Jordan curve on the 2-sphere by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously on the initial curve in Frechet distance in the case where the curve bisects the sphere. This even holds in the limit as time goes to infinity. This builds on Joseph Lauer's work on existence and uniqueness of solutions to the curvature flow problem on the sphere when the initial curve is not smooth.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"85 18","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1142/s1793525323500449
Genki Omori, Yuya Yoshida
The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We give a finite presentation for the balanced superelliptic handlebody group. To give this presentation, we construct a finite presentation for the liftable Hilden group.
{"title":"A Finite Presentation for the Balanced Superelliptic Handlebody Group","authors":"Genki Omori, Yuya Yoshida","doi":"10.1142/s1793525323500449","DOIUrl":"https://doi.org/10.1142/s1793525323500449","url":null,"abstract":"The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We give a finite presentation for the balanced superelliptic handlebody group. To give this presentation, we construct a finite presentation for the liftable Hilden group.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":" June","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1142/s1793525323500401
Kazuo Habiro, Anderson Vera
We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a good ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid $mathbb{N}^2$ of pairs on nonnegative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group $mathcal{M}$ of a surface $Sigma_{g,1}$ with one boundary component, equipped with the normal subgroups $bar{X}$, $bar{Y}$ of $pi_1(Sigma_{g,1})$ associated to a standard Heegaard splitting of the $3$-sphere. We also consider the case where the group $G$ is the automorphism group of a free group.
{"title":"Double Johnson Filtrations for Mapping Class Groups","authors":"Kazuo Habiro, Anderson Vera","doi":"10.1142/s1793525323500401","DOIUrl":"https://doi.org/10.1142/s1793525323500401","url":null,"abstract":"We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a good ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid $mathbb{N}^2$ of pairs on nonnegative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group $mathcal{M}$ of a surface $Sigma_{g,1}$ with one boundary component, equipped with the normal subgroups $bar{X}$, $bar{Y}$ of $pi_1(Sigma_{g,1})$ associated to a standard Heegaard splitting of the $3$-sphere. We also consider the case where the group $G$ is the automorphism group of a free group.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"84 14","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1142/s1793525323500450
Igor V. Nikolaev
{"title":"Remark on Ordered Braid Groups","authors":"Igor V. Nikolaev","doi":"10.1142/s1793525323500450","DOIUrl":"https://doi.org/10.1142/s1793525323500450","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":" 21","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1142/s1793525323500395
Elizabeth Gasparim, Luiz A. B. San Martin
We compare Lagrangian thimbles for the potential of a Landau-Ginzburg model to the Morse theory of its real part. We explore Landau-Ginzburg models defined using Lie theory, constructing their real Lagrangian thimbles explicitly and comparing them to the stable and unstable manifolds of the real gradient flow.
{"title":"Morse Functions and Real Lagrangian Thimbles on Adjoint Orbits","authors":"Elizabeth Gasparim, Luiz A. B. San Martin","doi":"10.1142/s1793525323500395","DOIUrl":"https://doi.org/10.1142/s1793525323500395","url":null,"abstract":"We compare Lagrangian thimbles for the potential of a Landau-Ginzburg model to the Morse theory of its real part. We explore Landau-Ginzburg models defined using Lie theory, constructing their real Lagrangian thimbles explicitly and comparing them to the stable and unstable manifolds of the real gradient flow.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"85 20","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-03DOI: 10.1142/s1793525323500292
Yasuaki Hiraoka, Yuichi Ike, Michio Yoshiwaki
We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander–Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan–Lesnick and the sheaf-theoretic convolution distance due to Kashiwara–Schapira.
{"title":"Algebraic stability theorem for derived categories of zigzag persistence modules","authors":"Yasuaki Hiraoka, Yuichi Ike, Michio Yoshiwaki","doi":"10.1142/s1793525323500292","DOIUrl":"https://doi.org/10.1142/s1793525323500292","url":null,"abstract":"We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander–Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan–Lesnick and the sheaf-theoretic convolution distance due to Kashiwara–Schapira.","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136228733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-24DOI: 10.1142/s1793525323920012
Huaxin Lin
{"title":"Erratum: Homomorphisms from AH-algebras","authors":"Huaxin Lin","doi":"10.1142/s1793525323920012","DOIUrl":"https://doi.org/10.1142/s1793525323920012","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"188 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74175252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-30DOI: 10.1142/s1793525323500310
Komal Negi, M. Prabhakar
{"title":"Numerical invariants for two-component virtual spatial graphs","authors":"Komal Negi, M. Prabhakar","doi":"10.1142/s1793525323500310","DOIUrl":"https://doi.org/10.1142/s1793525323500310","url":null,"abstract":"","PeriodicalId":49151,"journal":{"name":"Journal of Topology and Analysis","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79247240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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