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Polyhedral products in abstract and motivic homotopy theory 抽象和动机同调理论中的多面体积
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-19 DOI: arxiv-2406.13540
William Hornslien
{"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}
引用次数: 0
Intertwining category and complexity 类别与复杂性交织
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-18 DOI: arxiv-2406.12265
Ekansh Jauhari
{"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}
引用次数: 0
Stability of Persistent Path Diagrams 持久路径图的稳定性
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-17 DOI: arxiv-2406.11998
Shen Zhang
{"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}
引用次数: 0
On the splitting of surfaces in motivic stable homotopy category 论动机稳定同构范畴中的曲面分裂
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-17 DOI: arxiv-2406.11922
Haoyang Liu
{"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}
引用次数: 0
$ mathbb{Z}_{2} $- homology of the orbit spaces $ G_{n,2}/ T^{n} $ $ mathbb{Z}_{2} $- 轨道空间的同源性 $ G_{n,2}/ T^{n} $
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-17 DOI: arxiv-2406.11625
Vladimir Ivanović, Svjetlana Terzić
{"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}
引用次数: 0
Pro-nilpotently extended dgca-s and SH Lie-Rinehart pairs Pro-nilpotently extended dgca-s 和 SH Lie-Rinehart 对
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-16 DOI: arxiv-2406.10883
Damjan Pištalo
{"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}
引用次数: 0
On the extension problems for the 33-stem homotopy groups of the 6-, 7- and 8-spheres 关于 6、7 和 8 球体的 33 干同调群的扩展问题
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-12 DOI: arxiv-2406.08621
Juxin Yang, Jie Wu
{"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}
引用次数: 0
A topological analysis of the space of recipes 食谱空间的拓扑分析
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-12 DOI: arxiv-2406.09445
Emerson G. Escolar, Yuta Shimada, Masahiro Yuasa
{"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}
引用次数: 0
Smith homomorphisms and Spin$^h$ structures 斯密同态与 Spin$^h$ 结构
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-12 DOI: arxiv-2406.08237
Arun Debray, Cameron Krulewski
{"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}
引用次数: 0
Contractibility of Vietoris-Rips Complexes of dense subsets in $(mathbb{R}^n, ell_1)$ via hyperconvex embeddings 通过超凸嵌入看 $(mathbb{R}^n, ell_1)$ 中密集子集的 Vietoris-Rips 复合物的可收缩性
arXiv - MATH - Algebraic Topology Pub Date : 2024-06-12 DOI: arxiv-2406.08664
Qingsong Wang
{"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}
引用次数: 0
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