The purpose of this paper is to show how Positselski's relative nonhomogeneous Koszul duality theory applies when studying the linear category underlying the PROP associated to a (non-augmented) operad of a certain form, in particular assuming that the reduced part of the operad is binary quadratic. In this case, the linear category has both a left augmentation and a right augmentation (corresponding to different units), using Positselski's terminology. The general theory provides two associated linear differential graded (DG) categories; indeed, in this framework, one can work entirely within the DG realm, as opposed to the curved setting required for Positselski's general theory. Moreover, DG modules over DG categories are related by adjunctions. When the reduced part of the operad is Koszul (working over a field of characteristic zero), the relative Koszul duality theory shows that there is a Koszul-type equivalence between the appropriate homotopy categories of DG modules. This gives a form of Koszul duality relationship between the above DG categories. This is illustrated by the case of the operad encoding unital, commutative associative algebras, extending the classical Koszul duality between commutative associative algebras and Lie algebras. In this case, the associated linear category is the linearization of the category of finite sets and all maps. The relative nonhomogeneous Koszul duality theory relates its derived category to the respective homotopy categories of modules over two explicit linear DG categories.
{"title":"Relative nonhomogeneous Koszul duality for PROPs associated to nonaugmented operads","authors":"Geoffrey Powell","doi":"arxiv-2406.08132","DOIUrl":"https://doi.org/arxiv-2406.08132","url":null,"abstract":"The purpose of this paper is to show how Positselski's relative\u0000nonhomogeneous Koszul duality theory applies when studying the linear category\u0000underlying the PROP associated to a (non-augmented) operad of a certain form,\u0000in particular assuming that the reduced part of the operad is binary quadratic.\u0000In this case, the linear category has both a left augmentation and a right\u0000augmentation (corresponding to different units), using Positselski's\u0000terminology. The general theory provides two associated linear differential graded (DG)\u0000categories; indeed, in this framework, one can work entirely within the DG\u0000realm, as opposed to the curved setting required for Positselski's general\u0000theory. Moreover, DG modules over DG categories are related by adjunctions. When the reduced part of the operad is Koszul (working over a field of\u0000characteristic zero), the relative Koszul duality theory shows that there is a\u0000Koszul-type equivalence between the appropriate homotopy categories of DG\u0000modules. This gives a form of Koszul duality relationship between the above DG\u0000categories. This is illustrated by the case of the operad encoding unital, commutative\u0000associative algebras, extending the classical Koszul duality between\u0000commutative associative algebras and Lie algebras. In this case, the associated\u0000linear category is the linearization of the category of finite sets and all\u0000maps. The relative nonhomogeneous Koszul duality theory relates its derived\u0000category to the respective homotopy categories of modules over two explicit\u0000linear DG categories.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515043","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}
Soham Mukherjee, Shreyas N. Samaga, Cheng Xin, Steve Oudot, Tamal K. Dey
End-to-end topological learning using 1-parameter persistence is well-known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique called GRIL. We establish a theoretical foundation of differentiating GRIL producing D-GRIL. We show that D-GRIL can be used to learn a bifiltration function on standard benchmark graph datasets. Further, we exhibit that this framework can be applied in the context of bio-activity prediction in drug discovery.
{"title":"D-GRIL: End-to-End Topological Learning with 2-parameter Persistence","authors":"Soham Mukherjee, Shreyas N. Samaga, Cheng Xin, Steve Oudot, Tamal K. Dey","doi":"arxiv-2406.07100","DOIUrl":"https://doi.org/arxiv-2406.07100","url":null,"abstract":"End-to-end topological learning using 1-parameter persistence is well-known.\u0000We show that the framework can be enhanced using 2-parameter persistence by\u0000adopting a recently introduced 2-parameter persistence based vectorization\u0000technique called GRIL. We establish a theoretical foundation of differentiating\u0000GRIL producing D-GRIL. We show that D-GRIL can be used to learn a bifiltration\u0000function on standard benchmark graph datasets. Further, we exhibit that this\u0000framework can be applied in the context of bio-activity prediction in drug\u0000discovery.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508006","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 size of Sheehy's subdivision bifiltrations, up to homotopy. We focus in particular on the subdivision-Rips bifiltration $mathcal{SR}(X)$ of a metric space $X$, the only density-sensitive bifiltration on metric spaces known to satisfy a strong robustness property. Given a simplicial filtration $mathcal{F}$ with a total of $m$ maximal simplices across all indices, we introduce a nerve-based simplicial model for its subdivision bifiltration $mathcal{SF}$ whose $k$-skeleton has size $O(m^{k+1})$. We also show that the $0$-skeleton of any simplicial model of $mathcal{SF}$ has size at least $m$. We give several applications: For an arbitrary metric space $X$, we introduce a $sqrt{2}$-approximation to $mathcal{SR}(X)$, denoted $mathcal{J}(X)$, whose $k$-skeleton has size $O(|X|^{k+2})$. This improves on the previous best approximation bound of $sqrt{3}$, achieved by the degree-Rips bifiltration, which implies that $mathcal{J}(X)$ is more robust than degree-Rips. Moreover, we show that the approximation factor of $sqrt{2}$ is tight; in particular, there exists no exact model of $mathcal{SR}(X)$ with poly-size skeleta. On the other hand, we show that for $X$ in a fixed-dimensional Euclidean space with the $ell_p$-metric, there exists an exact model of $mathcal{SR}(X)$ with poly-size skeleta for $pin {1, infty}$, as well as a $(1+epsilon)$-approximation to $mathcal{SR}(X)$ with poly-size skeleta for any $p in (1, infty)$ and fixed ${epsilon > 0}$.
{"title":"Nerve Models of Subdivision Bifiltrations","authors":"Michael Lesnick, Ken McCabe","doi":"arxiv-2406.07679","DOIUrl":"https://doi.org/arxiv-2406.07679","url":null,"abstract":"We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We\u0000focus in particular on the subdivision-Rips bifiltration $mathcal{SR}(X)$ of a\u0000metric space $X$, the only density-sensitive bifiltration on metric spaces\u0000known to satisfy a strong robustness property. Given a simplicial filtration\u0000$mathcal{F}$ with a total of $m$ maximal simplices across all indices, we\u0000introduce a nerve-based simplicial model for its subdivision bifiltration\u0000$mathcal{SF}$ whose $k$-skeleton has size $O(m^{k+1})$. We also show that the\u0000$0$-skeleton of any simplicial model of $mathcal{SF}$ has size at least $m$.\u0000We give several applications: For an arbitrary metric space $X$, we introduce a\u0000$sqrt{2}$-approximation to $mathcal{SR}(X)$, denoted $mathcal{J}(X)$, whose\u0000$k$-skeleton has size $O(|X|^{k+2})$. This improves on the previous best\u0000approximation bound of $sqrt{3}$, achieved by the degree-Rips bifiltration,\u0000which implies that $mathcal{J}(X)$ is more robust than degree-Rips. Moreover,\u0000we show that the approximation factor of $sqrt{2}$ is tight; in particular,\u0000there exists no exact model of $mathcal{SR}(X)$ with poly-size skeleta. On the\u0000other hand, we show that for $X$ in a fixed-dimensional Euclidean space with\u0000the $ell_p$-metric, there exists an exact model of $mathcal{SR}(X)$ with\u0000poly-size skeleta for $pin {1, infty}$, as well as a\u0000$(1+epsilon)$-approximation to $mathcal{SR}(X)$ with poly-size skeleta for\u0000any $p in (1, infty)$ and fixed ${epsilon > 0}$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515044","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 this paper, we discuss the motivic stable homotopy type of abelian varieties. For an abelian variety over a field $k$ with a rational point, it always splits off a top-dimensional cell in motivic stable homotopy category $text{SH}(k)$. Let $k = mathbb{R}$, there is a concrete splitting which is determined by the motive of X and the real points $X(mathbb{R})$ in $text{SH}(mathbb{R})_mathbb{Q}$. We will also discuss this splitting from a viewpoint of the Chow-Witt correspondences.
本文讨论了无常变的动机稳定同调类型。对于一个有理点的域$k$上的无常变种，它在动机稳定同调类型$text{SH}(k)$中分裂出一个顶维单元。让 $k = mathbb{R}$，有一个具体的分裂，它是由 X 的动机和实点 $X(mathbb{R})$ 在$text{SH}(mathbb{R})_mathbb{Q}$中决定的。我们还将从周-维特对应关系的角度讨论这种分裂。
{"title":"Splitting of abelian varieties in motivic stable homotopy category","authors":"Haoyang Liu","doi":"arxiv-2406.05674","DOIUrl":"https://doi.org/arxiv-2406.05674","url":null,"abstract":"In this paper, we discuss the motivic stable homotopy type of abelian\u0000varieties. For an abelian variety over a field $k$ with a rational point, it\u0000always splits off a top-dimensional cell in motivic stable homotopy category\u0000$text{SH}(k)$. Let $k = mathbb{R}$, there is a concrete splitting which is\u0000determined by the motive of X and the real points $X(mathbb{R})$ in\u0000$text{SH}(mathbb{R})_mathbb{Q}$. We will also discuss this splitting from a\u0000viewpoint of the Chow-Witt correspondences.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515045","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}
Let $[SU(2n),mathscr{L}]$ denote the bordism class of $SU(2n)$ $(nge 2)$ equipped with the left invariant framing $mathscr{L}$. Then it is well known that $e_mathbb{C}([SU(2n), mathscr{L}])=0$ in $mathbb{O}/mathbb{Z}$ where $e_mathbb{C}$ denotes the complex Adams $e$-invariant. In this note we show that replacing $mathscr{L}$ by the twisted framing by a specific map it can be transformed into a generator of $mathrm{Im} , e_mathbb{C}$. In addition to that we also show that the same procedure affords an analogous result for a quotient of $SU(2n+1)$ by a circle subgroup which inherits a canonical framing from $SU(2n+1)$ in the usual way.
{"title":"The special unitary groups $SU(2n)$ as framed manifolds","authors":"Haruo Minami","doi":"arxiv-2406.11878","DOIUrl":"https://doi.org/arxiv-2406.11878","url":null,"abstract":"Let $[SU(2n),mathscr{L}]$ denote the bordism class of $SU(2n)$ $(nge 2)$\u0000equipped with the left invariant framing $mathscr{L}$. Then it is well known\u0000that $e_mathbb{C}([SU(2n), mathscr{L}])=0$ in $mathbb{O}/mathbb{Z}$ where\u0000$e_mathbb{C}$ denotes the complex Adams $e$-invariant. In this note we show\u0000that replacing $mathscr{L}$ by the twisted framing by a specific map it can be\u0000transformed into a generator of $mathrm{Im} , e_mathbb{C}$. In addition to\u0000that we also show that the same procedure affords an analogous result for a\u0000quotient of $SU(2n+1)$ by a circle subgroup which inherits a canonical framing\u0000from $SU(2n+1)$ in the usual way.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515047","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}
Several methods have been proposed to define tangent spaces for diffeological spaces. Among them, the internal tangent functor is obtained as the left Kan extension of the tangent functor for manifolds. However, the right Kan extension of the same functor has not been well-studied. In this paper, we investigate the relationship between this right Kan extension and the external tangent space, another type of tangent space for diffeological spaces. We prove that by slightly modifying the inclusion functor used in the right Kan extension, we obtain a right tangent space functor, which is almost isomorphic to the external tangent space. Furthermore, we show that when a diffeological space satisfies a favorable property called smoothly regular, this right tangent space coincides with the right Kan extension mentioned earlier.
{"title":"Tangent spaces of diffeological spaces and their variants","authors":"Masaki Taho","doi":"arxiv-2406.04703","DOIUrl":"https://doi.org/arxiv-2406.04703","url":null,"abstract":"Several methods have been proposed to define tangent spaces for diffeological\u0000spaces. Among them, the internal tangent functor is obtained as the left Kan\u0000extension of the tangent functor for manifolds. However, the right Kan\u0000extension of the same functor has not been well-studied. In this paper, we\u0000investigate the relationship between this right Kan extension and the external\u0000tangent space, another type of tangent space for diffeological spaces. We prove\u0000that by slightly modifying the inclusion functor used in the right Kan\u0000extension, we obtain a right tangent space functor, which is almost isomorphic\u0000to the external tangent space. Furthermore, we show that when a diffeological\u0000space satisfies a favorable property called smoothly regular, this right\u0000tangent space coincides with the right Kan extension mentioned earlier.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508008","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}
Graph structures offer a versatile framework for representing diverse patterns in nature and complex systems, applicable across domains like molecular chemistry, social networks, and transportation systems. While diffusion models have excelled in generating various objects, generating graphs remains challenging. This thesis explores the potential of score-based generative models in generating such objects through a modelization as combinatorial complexes, which are powerful topological structures that encompass higher-order relationships. In this thesis, we propose a unified framework by employing stochastic differential equations. We not only generalize the generation of complex objects such as graphs and hypergraphs, but we also unify existing generative modelling approaches such as Score Matching with Langevin dynamics and Denoising Diffusion Probabilistic Models. This innovation overcomes limitations in existing frameworks that focus solely on graph generation, opening up new possibilities in generative AI. The experiment results showed that our framework could generate these complex objects, and could also compete against state-of-the-art approaches for mere graph and molecule generation tasks.
{"title":"Combinatorial Complex Score-based Diffusion Modelling through Stochastic Differential Equations","authors":"Adrien Carrel","doi":"arxiv-2406.04916","DOIUrl":"https://doi.org/arxiv-2406.04916","url":null,"abstract":"Graph structures offer a versatile framework for representing diverse\u0000patterns in nature and complex systems, applicable across domains like\u0000molecular chemistry, social networks, and transportation systems. While\u0000diffusion models have excelled in generating various objects, generating graphs\u0000remains challenging. This thesis explores the potential of score-based\u0000generative models in generating such objects through a modelization as\u0000combinatorial complexes, which are powerful topological structures that\u0000encompass higher-order relationships. In this thesis, we propose a unified framework by employing stochastic\u0000differential equations. We not only generalize the generation of complex\u0000objects such as graphs and hypergraphs, but we also unify existing generative\u0000modelling approaches such as Score Matching with Langevin dynamics and\u0000Denoising Diffusion Probabilistic Models. This innovation overcomes limitations\u0000in existing frameworks that focus solely on graph generation, opening up new\u0000possibilities in generative AI. The experiment results showed that our framework could generate these complex\u0000objects, and could also compete against state-of-the-art approaches for mere\u0000graph and molecule generation tasks.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515046","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}
Solid modules over $mathbb{Q}$ or $mathbb{F}_p$, introduced by Clausen and Scholze, are a well-behaved variant of complete topological vector spaces that forms a symmetric monoidal Grothendieck abelian category. For a discrete field $k$, we construct the category of ultrasolid $k$-modules, which specialises to solid modules over $mathbb{Q}$ or $mathbb{F}_p$. In this setting, we show some commutative algebra results like an ultrasolid variant of Nakayama's lemma. We also explore higher algebra in the form of animated and $mathbb{E}_infty$ ultrasolid $k$-algebras, and their deformation theory. We focus on the subcategory of complete profinite $k$-algebras, which we prove is contravariantly equivalent to equal characteristic formal moduli problems with coconnective tangent complex.
{"title":"Ultrasolid Homotopical Algebra","authors":"Sofía Marlasca Aparicio","doi":"arxiv-2406.04063","DOIUrl":"https://doi.org/arxiv-2406.04063","url":null,"abstract":"Solid modules over $mathbb{Q}$ or $mathbb{F}_p$, introduced by Clausen and\u0000Scholze, are a well-behaved variant of complete topological vector spaces that\u0000forms a symmetric monoidal Grothendieck abelian category. For a discrete field\u0000$k$, we construct the category of ultrasolid $k$-modules, which specialises to\u0000solid modules over $mathbb{Q}$ or $mathbb{F}_p$. In this setting, we show\u0000some commutative algebra results like an ultrasolid variant of Nakayama's\u0000lemma. We also explore higher algebra in the form of animated and\u0000$mathbb{E}_infty$ ultrasolid $k$-algebras, and their deformation theory. We\u0000focus on the subcategory of complete profinite $k$-algebras, which we prove is\u0000contravariantly equivalent to equal characteristic formal moduli problems with\u0000coconnective tangent complex.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515048","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}
Let $mathcal{X}$ be a smooth Fano threefold over the complex numbers of Picard rank $1$ with finite automorphism group. We give numerical restrictions on the order of the automorphism group $mathrm{Aut}(mathcal{X})$ provided the genus $g(mathcal{X})leq 10$ and $mathcal{X}$ is not an ordinary smooth Gushel-Mukai threefold. More precisely, we show that the order $|mathrm{Aut}(mathcal{X})|$ divides a certain explicit number depending on the genus of $mathcal{X}$. We use a classification of Fano threefolds in terms of complete intersections in homogeneous varieties and the previous paper of A. Gorinov and the author regarding the topology of spaces of regular sections.
{"title":"On the automorphism groups of smooth Fano threefolds","authors":"Nikolay Konovalov","doi":"arxiv-2406.03584","DOIUrl":"https://doi.org/arxiv-2406.03584","url":null,"abstract":"Let $mathcal{X}$ be a smooth Fano threefold over the complex numbers of\u0000Picard rank $1$ with finite automorphism group. We give numerical restrictions\u0000on the order of the automorphism group $mathrm{Aut}(mathcal{X})$ provided the\u0000genus $g(mathcal{X})leq 10$ and $mathcal{X}$ is not an ordinary smooth\u0000Gushel-Mukai threefold. More precisely, we show that the order\u0000$|mathrm{Aut}(mathcal{X})|$ divides a certain explicit number depending on\u0000the genus of $mathcal{X}$. We use a classification of Fano threefolds in terms\u0000of complete intersections in homogeneous varieties and the previous paper of A.\u0000Gorinov and the author regarding the topology of spaces of regular sections.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515049","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 describe the relation of $r$-similarity and finite-order invariants on the homotopy set $[S^1,Y]=pi_1(Y)$.
我们描述了同调集$[S^1,Y]=pi_1(Y)$上的$r$相似性与有限阶不变式的关系。
{"title":"Homotopy similarity of maps. Maps of the circle","authors":"S. S. Podkorytov","doi":"arxiv-2406.02526","DOIUrl":"https://doi.org/arxiv-2406.02526","url":null,"abstract":"We describe the relation of $r$-similarity and finite-order invariants on the\u0000homotopy set $[S^1,Y]=pi_1(Y)$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257330","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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