We introduce polyhedral products in an $infty$-categorical setting. We generalize a splitting result by Bahri, Bendersky, Cohen, and Gitler that determines the stable homotopy type of the a polyhedral product. We also introduce a motivic refinement of moment-angle complexes and use the splitting result to compute cellular $mathbb{A}^1$-homology, and $mathbb{A}^1$-Euler characteristics.
{"title":"Polyhedral products in abstract and motivic homotopy theory","authors":"William Hornslien","doi":"arxiv-2406.13540","DOIUrl":"https://doi.org/arxiv-2406.13540","url":null,"abstract":"We introduce polyhedral products in an $infty$-categorical setting. We\u0000generalize a splitting result by Bahri, Bendersky, Cohen, and Gitler that\u0000determines the stable homotopy type of the a polyhedral product. We also\u0000introduce a motivic refinement of moment-angle complexes and use the splitting\u0000result to compute cellular $mathbb{A}^1$-homology, and $mathbb{A}^1$-Euler\u0000characteristics.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515034","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 develop the theory of the intertwining distributional versions of the LS-category and the sequential topological complexities of a space $X$, denoted by $imathsf{cat}(X)$ and $imathsf{TC}_m(X)$, respectively. We prove that they satisfy most of the nice properties as their respective distributional counterparts $dmathsf{cat}(X)$ and $dmathsf{TC}_m(X)$, and their classical counterparts $mathsf{cat}(X)$ and $mathsf{TC}_m(X)$, such as homotopy invariance and special behavior on topological groups. We show that the notions of $imathsf{TC}_m$ and $dmathsf{TC}_m$ are different for each $m ge 2$ by proving that $imathsf{TC}_m(mathcal{H})=1$ for all $m ge 2$ for Higman's group $mathcal{H}$. Using cohomological lower bounds, we also provide various examples of locally finite CW complexes $X$ for which $imathsf{cat}(X) > 1$, $imathsf{TC}_m(X) > 1$, $imathsf{cat}(X) = dmathsf{cat}(X) = mathsf{cat}(X)$, and $imathsf{TC}(X) = dmathsf{TC}(X) = mathsf{TC}(X)$.
{"title":"Intertwining category and complexity","authors":"Ekansh Jauhari","doi":"arxiv-2406.12265","DOIUrl":"https://doi.org/arxiv-2406.12265","url":null,"abstract":"We develop the theory of the intertwining distributional versions of the\u0000LS-category and the sequential topological complexities of a space $X$, denoted\u0000by $imathsf{cat}(X)$ and $imathsf{TC}_m(X)$, respectively. We prove that they\u0000satisfy most of the nice properties as their respective distributional\u0000counterparts $dmathsf{cat}(X)$ and $dmathsf{TC}_m(X)$, and their classical\u0000counterparts $mathsf{cat}(X)$ and $mathsf{TC}_m(X)$, such as homotopy\u0000invariance and special behavior on topological groups. We show that the notions\u0000of $imathsf{TC}_m$ and $dmathsf{TC}_m$ are different for each $m ge 2$ by\u0000proving that $imathsf{TC}_m(mathcal{H})=1$ for all $m ge 2$ for Higman's\u0000group $mathcal{H}$. Using cohomological lower bounds, we also provide various\u0000examples of locally finite CW complexes $X$ for which $imathsf{cat}(X) > 1$,\u0000$imathsf{TC}_m(X) > 1$, $imathsf{cat}(X) = dmathsf{cat}(X) =\u0000mathsf{cat}(X)$, and $imathsf{TC}(X) = dmathsf{TC}(X) = mathsf{TC}(X)$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515035","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 real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections between nodes. With the deepening of research, networks have been endowed with richer structures, such as directed edges, edge weights, and even hyperedges involving multiple nodes. Persistent homology is an algebraic method for analyzing data. It helps us understand the intrinsic structure and patterns of data by tracking the death and birth of topological features at different scale parameters.The original persistent homology is not suitable for directed networks. However, the introduction of path homology established on digraphs solves this problem. This paper studies complex networks represented as weighted digraphs or edge-weighted path complexes and their persistent path homology. We use the homotopy theory of digraphs and path complexes, along with the interleaving property of persistent modules and bottleneck distance, to prove the stability of persistent path diagram with respect to weighted digraphs or edge-weighted path complexes. Therefore, persistent path homology has practical application value.
{"title":"Stability of Persistent Path Diagrams","authors":"Shen Zhang","doi":"arxiv-2406.11998","DOIUrl":"https://doi.org/arxiv-2406.11998","url":null,"abstract":"In real-world systems, the relationships and connections between components\u0000are highly complex. Real systems are often described as networks, where nodes\u0000represent objects in the system and edges represent relationships or\u0000connections between nodes. With the deepening of research, networks have been\u0000endowed with richer structures, such as directed edges, edge weights, and even\u0000hyperedges involving multiple nodes. Persistent homology is an algebraic method for analyzing data. It helps us\u0000understand the intrinsic structure and patterns of data by tracking the death\u0000and birth of topological features at different scale parameters.The original\u0000persistent homology is not suitable for directed networks. However, the\u0000introduction of path homology established on digraphs solves this problem. This\u0000paper studies complex networks represented as weighted digraphs or\u0000edge-weighted path complexes and their persistent path homology. We use the\u0000homotopy theory of digraphs and path complexes, along with the interleaving\u0000property of persistent modules and bottleneck distance, to prove the stability\u0000of persistent path diagram with respect to weighted digraphs or edge-weighted\u0000path complexes. Therefore, persistent path homology has practical application\u0000value.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515036","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 $k$ be a field and $X$ be a smooth projective surface over $k$ with a rational point, we discuss the condition of splitting off the top cell for the motivic stable homotopy type of $X$. We also study some outlying examples, such as K3 surfaces.
{"title":"On the splitting of surfaces in motivic stable homotopy category","authors":"Haoyang Liu","doi":"arxiv-2406.11922","DOIUrl":"https://doi.org/arxiv-2406.11922","url":null,"abstract":"Let $k$ be a field and $X$ be a smooth projective surface over $k$ with a\u0000rational point, we discuss the condition of splitting off the top cell for the\u0000motivic stable homotopy type of $X$. We also study some outlying examples, such\u0000as K3 surfaces.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"189 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515038","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 $mathbb{Z}_2$-homology groups of the orbit space $X_n = G_{n,2}/T^n$ for the canonical action of the compact torus $T^n$ on a complex Grassmann manifold $G_{n,2}$. Our starting point is the model $(U_n, p_n)$ for $X_n$ constructed by Buchstaber and Terzi'c (2020), where $U_n = Delta _{n,2}times mathcal{F}_{n}$ for a hypersimplex $Delta_{n,2}$ and an universal space of parameters $mathcal{F}_{n}$ defined in Buchstaber and Terzi'c (2019), (2020). It is proved by Buchstaber and Terzi'c (2021) that $mathcal{F}_{n}$ is diffeomorphic to the moduli space $mathcal{M}_{0,n}$ of stable $n$-pointed genus zero curves. We exploit the results from Keel (1992) and Ceyhan (2009) on homology groups of $mathcal{M}_{0,n}$ and express them in terms of the stratification of $mathcal{F}_{n}$ which are incorporated in the model $(U_n, p_n)$. In the result we provide the description of cycles in $X_n$, inductively on $ n. $ We obtain as well explicit formulas for $mathbb{Z}_2$-homology groups for $X_5$ and $X_6$. The results for $X_5$ recover by different method the results from Buchstaber and Terzi'c (2021) and S"uss (2020). The results for $X_6$ we consider to be new.
{"title":"$ mathbb{Z}_{2} $- homology of the orbit spaces $ G_{n,2}/ T^{n} $","authors":"Vladimir Ivanović, Svjetlana Terzić","doi":"arxiv-2406.11625","DOIUrl":"https://doi.org/arxiv-2406.11625","url":null,"abstract":"We study the $mathbb{Z}_2$-homology groups of the orbit space $X_n =\u0000G_{n,2}/T^n$ for the canonical action of the compact torus $T^n$ on a complex\u0000Grassmann manifold $G_{n,2}$. Our starting point is the model $(U_n, p_n)$ for\u0000$X_n$ constructed by Buchstaber and Terzi'c (2020), where $U_n = Delta\u0000_{n,2}times mathcal{F}_{n}$ for a hypersimplex $Delta_{n,2}$ and an\u0000universal space of parameters $mathcal{F}_{n}$ defined in Buchstaber and\u0000Terzi'c (2019), (2020). It is proved by Buchstaber and Terzi'c (2021) that\u0000$mathcal{F}_{n}$ is diffeomorphic to the moduli space $mathcal{M}_{0,n}$ of\u0000stable $n$-pointed genus zero curves. We exploit the results from Keel (1992)\u0000and Ceyhan (2009) on homology groups of $mathcal{M}_{0,n}$ and express them in\u0000terms of the stratification of $mathcal{F}_{n}$ which are incorporated in the\u0000model $(U_n, p_n)$. In the result we provide the description of cycles in\u0000$X_n$, inductively on $ n. $ We obtain as well explicit formulas for\u0000$mathbb{Z}_2$-homology groups for $X_5$ and $X_6$. The results for $X_5$\u0000recover by different method the results from Buchstaber and Terzi'c (2021) and\u0000S\"uss (2020). The results for $X_6$ we consider to be new.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515037","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}
Category of pro-nilpotently extended differential graded commutative algebras is introduced. Chevalley-Eilenberg construction provides an equivalence between its certain full subcategory and the opposite to the full subcategory of strong homotopy Lie Rinehart pairs with strong homotopy morphisms, consisting of pairs $(A,M)$ where $M$ is flat as a graded $A$-module. It is shown that pairs $(A,M)$, where $A$ is a semi-free dgca and $M$ a cell complex in $op{Mod}(A)$, form a category of fibrant objects by proving that their Chevalley-Eilenberg complexes form a category of cofibrant objects.
{"title":"Pro-nilpotently extended dgca-s and SH Lie-Rinehart pairs","authors":"Damjan Pištalo","doi":"arxiv-2406.10883","DOIUrl":"https://doi.org/arxiv-2406.10883","url":null,"abstract":"Category of pro-nilpotently extended differential graded commutative algebras\u0000is introduced. Chevalley-Eilenberg construction provides an equivalence between\u0000its certain full subcategory and the opposite to the full subcategory of strong\u0000homotopy Lie Rinehart pairs with strong homotopy morphisms, consisting of pairs\u0000$(A,M)$ where $M$ is flat as a graded $A$-module. It is shown that pairs\u0000$(A,M)$, where $A$ is a semi-free dgca and $M$ a cell complex in $op{Mod}(A)$,\u0000form a category of fibrant objects by proving that their Chevalley-Eilenberg\u0000complexes form a category of cofibrant objects.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"186 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508007","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}
This paper tackles the extension problems for the homotopy groups $pi_{39}(S^{6})$, $pi_{40}(S^{7})$, and $pi_{41}(S^{8})$ localized at 2, the puzzles having remained unsolved for forty-five years. We introduce a tool for the theory of determinations of unstable homotopy groups, namely, the $mathcal{Z}$-shape Toda bracket, by which we are able to solve the extension problems with respect to these three homotopy groups.
{"title":"On the extension problems for the 33-stem homotopy groups of the 6-, 7- and 8-spheres","authors":"Juxin Yang, Jie Wu","doi":"arxiv-2406.08621","DOIUrl":"https://doi.org/arxiv-2406.08621","url":null,"abstract":"This paper tackles the extension problems for the homotopy groups\u0000$pi_{39}(S^{6})$, $pi_{40}(S^{7})$, and $pi_{41}(S^{8})$ localized at 2, the\u0000puzzles having remained unsolved for forty-five years. We introduce a tool for\u0000the theory of determinations of unstable homotopy groups, namely, the\u0000$mathcal{Z}$-shape Toda bracket, by which we are able to solve the extension\u0000problems with respect to these three homotopy groups.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515041","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 recent years, the use of data-driven methods has provided insights into underlying patterns and principles behind culinary recipes. In this exploratory work, we introduce the use of topological data analysis, especially persistent homology, in order to study the space of culinary recipes. In particular, persistent homology analysis provides a set of recipes surrounding the multiscale "holes" in the space of existing recipes. We then propose a method to generate novel ingredient combinations using combinatorial optimization on this topological information. We made biscuits using the novel ingredient combinations, which were confirmed to be acceptable enough by a sensory evaluation study. Our findings indicate that topological data analysis has the potential for providing new tools and insights in the study of culinary recipes.
{"title":"A topological analysis of the space of recipes","authors":"Emerson G. Escolar, Yuta Shimada, Masahiro Yuasa","doi":"arxiv-2406.09445","DOIUrl":"https://doi.org/arxiv-2406.09445","url":null,"abstract":"In recent years, the use of data-driven methods has provided insights into\u0000underlying patterns and principles behind culinary recipes. In this exploratory\u0000work, we introduce the use of topological data analysis, especially persistent\u0000homology, in order to study the space of culinary recipes. In particular,\u0000persistent homology analysis provides a set of recipes surrounding the\u0000multiscale \"holes\" in the space of existing recipes. We then propose a method\u0000to generate novel ingredient combinations using combinatorial optimization on\u0000this topological information. We made biscuits using the novel ingredient\u0000combinations, which were confirmed to be acceptable enough by a sensory\u0000evaluation study. Our findings indicate that topological data analysis has the\u0000potential for providing new tools and insights in the study of culinary\u0000recipes.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"173 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515039","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 article, we answer two questions of Buchanan-McKean (arXiv:2312.08209) about bordism for manifolds with spin$^h$ structures: we establish a Smith isomorphism between the reduced spin$^h$ bordism of $mathbb{RP}^infty$ and pin$^{h-}$ bordism, and we provide a geometric explanation for the isomorphism $Omega_{4k}^{mathrm{Spin}^c} otimesmathbb Z[1/2] cong Omega_{4k}^{mathrm{Spin}^h} otimesmathbb Z[1/2]$. Our proofs use the general theory of twisted spin structures and Smith homomorphisms that we developed in arXiv:2405.04649 joint with Devalapurkar, Liu, Pacheco-Tallaj, and Thorngren, specifically that the Smith homomorphism participates in a long exact sequence with explicit, computable terms.
{"title":"Smith homomorphisms and Spin$^h$ structures","authors":"Arun Debray, Cameron Krulewski","doi":"arxiv-2406.08237","DOIUrl":"https://doi.org/arxiv-2406.08237","url":null,"abstract":"In this article, we answer two questions of Buchanan-McKean\u0000(arXiv:2312.08209) about bordism for manifolds with spin$^h$ structures: we\u0000establish a Smith isomorphism between the reduced spin$^h$ bordism of\u0000$mathbb{RP}^infty$ and pin$^{h-}$ bordism, and we provide a geometric\u0000explanation for the isomorphism $Omega_{4k}^{mathrm{Spin}^c} otimesmathbb\u0000Z[1/2] cong Omega_{4k}^{mathrm{Spin}^h} otimesmathbb Z[1/2]$. Our proofs\u0000use the general theory of twisted spin structures and Smith homomorphisms that\u0000we developed in arXiv:2405.04649 joint with Devalapurkar, Liu, Pacheco-Tallaj,\u0000and Thorngren, specifically that the Smith homomorphism participates in a long\u0000exact sequence with explicit, computable terms.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515042","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 consider the contractibility of Vietoris-Rips complexes of dense subsets of $(mathbb{R}^n,ell_1)$ with sufficiently large scales. This is motivated by a question by Matthew Zaremsky regarding whether for each $n$ natural there is a $r_n>0$ so that the Vietoris-Rips complex of $(mathbb{Z}^n,ell_1)$ at scale $r$ is contractible for all $rgeq r_n$. We approach this question using results that relates to the neighborhood of embeddings into hyperconvex metric space of a metric space $X$ and its connection to the Vietoris-Rips complex of $X$. In this manner, we provide positive answers to the question above for the case $n=2$ and $3$.
{"title":"Contractibility of Vietoris-Rips Complexes of dense subsets in $(mathbb{R}^n, ell_1)$ via hyperconvex embeddings","authors":"Qingsong Wang","doi":"arxiv-2406.08664","DOIUrl":"https://doi.org/arxiv-2406.08664","url":null,"abstract":"We consider the contractibility of Vietoris-Rips complexes of dense subsets\u0000of $(mathbb{R}^n,ell_1)$ with sufficiently large scales. This is motivated by\u0000a question by Matthew Zaremsky regarding whether for each $n$ natural there is\u0000a $r_n>0$ so that the Vietoris-Rips complex of $(mathbb{Z}^n,ell_1)$ at scale\u0000$r$ is contractible for all $rgeq r_n$. We approach this question using\u0000results that relates to the neighborhood of embeddings into hyperconvex metric\u0000space of a metric space $X$ and its connection to the Vietoris-Rips complex of\u0000$X$. In this manner, we provide positive answers to the question above for the\u0000case $n=2$ and $3$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515040","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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