Let $mathbb S^{infty}/mathbb Z_2$ be the infinite lens space. Denote the Steenrod algebra over the prime field $mathbb F_2$ by $mathscr A.$ It is well-known that the cohomology $H^{*}((mathbb S^{infty}/mathbb Z_2)^{oplus s}; mathbb F_2)$ is the polynomial algebra $mathcal {P}_s:= mathbb F_2[x_1, ldots, x_s],, deg(x_i) = 1,, i = 1,, 2,ldots, s.$ The Kameko squaring operation $(widetilde {Sq^0_*})_{(s; N)}: (mathbb F_2otimes_{mathscr A} mathcal {P}_s)_{2N+s} longrightarrow (mathbb F_2otimes_{mathscr A} mathcal {P}_s)_{N}$ is indeed a valuable homomorphism for studying the dimension of the indecomposables $mathbb F_2otimes_{mathscr A} mathcal {P}_s,$ It has been demonstrated that this $(widetilde {Sq^0_*})_{(s; N)}$ is onto. Motivated by our previous work [J. Korean Math. Soc. textbf{58} (2021), 643-702], this paper studies the kernel of the Kameko $(widetilde {Sq^0_*})_{(s; N_d)}$ for the case where $s = 5$ and the generic degree $N_d = 5(2^{d} - 1) + 11.2^{d+1}.$ We then rectify almost all of the main results that were incorrect in Nguyen Khac Tin's paper [Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. textbf{116}:81 (2022)]. We have also constructed several advanced algorithms in SAGEMATH to validate our results. These new algorithms make an important contribution to tackling the intricate task of explicitly determining both the dimension and the basis for the indecomposables $mathbb F_2 otimes_{mathscr A} mathcal {P}_s$ at positive degrees, a problem concerning algorithmic approaches that had not previously been addressed by any author. Furthermore, this paper encompasses an investigation of the fifth cohomological transfer's behavior in the aforementioned degrees $N_d.$
{"title":"On the mod-2 cohomology of the product of the infinite lens space and the space of invariants in a generic degree","authors":"Dang Vo Phuc","doi":"arxiv-2408.07485","DOIUrl":"https://doi.org/arxiv-2408.07485","url":null,"abstract":"Let $mathbb S^{infty}/mathbb Z_2$ be the infinite lens space. Denote the\u0000Steenrod algebra over the prime field $mathbb F_2$ by $mathscr A.$ It is\u0000well-known that the cohomology $H^{*}((mathbb S^{infty}/mathbb Z_2)^{oplus\u0000s}; mathbb F_2)$ is the polynomial algebra $mathcal {P}_s:= mathbb F_2[x_1,\u0000ldots, x_s],, deg(x_i) = 1,, i = 1,, 2,ldots, s.$ The Kameko squaring\u0000operation $(widetilde {Sq^0_*})_{(s; N)}: (mathbb F_2otimes_{mathscr A}\u0000mathcal {P}_s)_{2N+s} longrightarrow (mathbb F_2otimes_{mathscr A}\u0000mathcal {P}_s)_{N}$ is indeed a valuable homomorphism for studying the\u0000dimension of the indecomposables $mathbb F_2otimes_{mathscr A} mathcal\u0000{P}_s,$ It has been demonstrated that this $(widetilde {Sq^0_*})_{(s; N)}$ is\u0000onto. Motivated by our previous work [J. Korean Math. Soc. textbf{58} (2021),\u0000643-702], this paper studies the kernel of the Kameko $(widetilde\u0000{Sq^0_*})_{(s; N_d)}$ for the case where $s = 5$ and the generic degree $N_d =\u00005(2^{d} - 1) + 11.2^{d+1}.$ We then rectify almost all of the main results that\u0000were incorrect in Nguyen Khac Tin's paper [Rev. Real Acad. Cienc. Exactas Fis.\u0000Nat. Ser. A-Mat. textbf{116}:81 (2022)]. We have also constructed several\u0000advanced algorithms in SAGEMATH to validate our results. These new algorithms\u0000make an important contribution to tackling the intricate task of explicitly\u0000determining both the dimension and the basis for the indecomposables $mathbb\u0000F_2 otimes_{mathscr A} mathcal {P}_s$ at positive degrees, a problem\u0000concerning algorithmic approaches that had not previously been addressed by any\u0000author. Furthermore, this paper encompasses an investigation of the fifth\u0000cohomological transfer's behavior in the aforementioned degrees $N_d.$","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195599","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}
Write $P_k:= mathbb F_2[x_1,x_2,ldots ,x_k]$ for the polynomial algebra over the prime field $mathbb F_2$ with two elements, in $k$ generators $x_1, x_2, ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra, $mathcal A$. Let $GL_k$ be the general linear group over the field $mathbb F_2$. This group acts naturally on $P_k$ by matrix substitution. Since the two actions of $mathcal A$ and $GL_k$ upon $P_k$ commute with each other, there is an inherit action of $GL_k$ on $mathbb F_2{otimes}_{mathcal A}P_k$. Denote by $(mathbb F_2{otimes}_{mathcal A}P_k)_n^{GL_k}$ the subspace of $mathbb F_2{otimes}_{mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $n$. In 1989, Singer [23] defined the homological algebraic transfer $$varphi_k :mbox{Tor}^{mathcal A}_{k,n+k}(mathbb F_2,mathbb F_2) longrightarrow (mathbb F_2{otimes}_{mathcal A}P_k)_n^{GL_k},$$ where $mbox{Tor}^{mathcal{A}}_{k, k+n}(mathbb{F}_2, mathbb{F}_2)$ is the dual of Ext$_{mathcal{A}}^{k,k+n}(mathbb F_2,mathbb F_2)$, the $E_2$ term of the Adams spectral sequence of spheres. In general, the transfer $varphi_k$ is not a monomorphism and Singer made a conjecture that $varphi_k$ is an epimorphism for any $k geqslant 0$. The conjecture is studied by many authors. It is true for $k leqslant 3$ but unknown for $k geqslant 4$. In this paper, by using a technique of the Peterson hit problem we prove that Singer's conjecture is not true for $k=5$ and the internal degree $n = 108$. This result also refutes a one of Ph'uc in [19].
{"title":"A counter-example to Singer's conjecture for the algebraic transfer","authors":"Nguyen Sum","doi":"arxiv-2408.06669","DOIUrl":"https://doi.org/arxiv-2408.06669","url":null,"abstract":"Write $P_k:= mathbb F_2[x_1,x_2,ldots ,x_k]$ for the polynomial algebra\u0000over the prime field $mathbb F_2$ with two elements, in $k$ generators $x_1,\u0000x_2, ldots , x_k$, each of degree 1. The polynomial algebra $P_k$ is\u0000considered as a module over the mod-2 Steenrod algebra, $mathcal A$. Let\u0000$GL_k$ be the general linear group over the field $mathbb F_2$. This group\u0000acts naturally on $P_k$ by matrix substitution. Since the two actions of\u0000$mathcal A$ and $GL_k$ upon $P_k$ commute with each other, there is an inherit\u0000action of $GL_k$ on $mathbb F_2{otimes}_{mathcal A}P_k$. Denote by $(mathbb\u0000F_2{otimes}_{mathcal A}P_k)_n^{GL_k}$ the subspace of $mathbb\u0000F_2{otimes}_{mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of\u0000degree $n$. In 1989, Singer [23] defined the homological algebraic transfer\u0000$$varphi_k :mbox{Tor}^{mathcal A}_{k,n+k}(mathbb F_2,mathbb F_2)\u0000longrightarrow (mathbb F_2{otimes}_{mathcal A}P_k)_n^{GL_k},$$ where\u0000$mbox{Tor}^{mathcal{A}}_{k, k+n}(mathbb{F}_2, mathbb{F}_2)$ is the dual of\u0000Ext$_{mathcal{A}}^{k,k+n}(mathbb F_2,mathbb F_2)$, the $E_2$ term of the\u0000Adams spectral sequence of spheres. In general, the transfer $varphi_k$ is not\u0000a monomorphism and Singer made a conjecture that $varphi_k$ is an epimorphism\u0000for any $k geqslant 0$. The conjecture is studied by many authors. It is true\u0000for $k leqslant 3$ but unknown for $k geqslant 4$. In this paper, by using a\u0000technique of the Peterson hit problem we prove that Singer's conjecture is not\u0000true for $k=5$ and the internal degree $n = 108$. This result also refutes a\u0000one of Ph'uc in [19].","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195604","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}
For every stable presentably symmetric monoidal $infty$-category $mathcal{C}$ we use the Koszul duality between the spectral Lie operad and the cocommutative cooperad to construct an enveloping Hopf algebra functor $mathcal{U}: mathrm{Alg}_{mathrm{Lie}}(mathcal{C}) to mathrm{Hopf}(mathcal{C})$ from spectral Lie algebras in $mathcal{C}$ to cocommutative Hopf algebras in $mathcal{C}$ left adjoint to a functor of derived primitive elements. We prove that if $mathcal{C}$ is a rational stable presentably symmetric monoidal $infty$-category, the enveloping Hopf algebra functor is fully faithful. We conclude that Lie algebras in $mathcal{C}$ are algebras over the monad underlying the adjunction $T simeq mathcal{U} circ mathrm{Lie}: mathcal{C} rightleftarrows mathrm{Alg}_{mathrm{Lie}}(mathcal{C}) to mathrm{Hopf}(mathcal{C}), $ where $mathrm{Lie}$ is the free Lie algebra and $mathrm{T}$ is the tensor algebra. For general $mathcal{C}$ we introduce the notion of restricted $L_infty$-algebra as an algebra over the latter adjunction. For any field $K$ we construct a forgetful functor from restricted Lie algebras in connective $H(K)$-modules to the $infty$-category underlying a right induced model structure on simplicial restricted Lie algebras over $K $.
{"title":"Restricted $L_infty$-algebras and a derived Milnor-Moore theorem","authors":"Hadrian Heine","doi":"arxiv-2408.06917","DOIUrl":"https://doi.org/arxiv-2408.06917","url":null,"abstract":"For every stable presentably symmetric monoidal $infty$-category\u0000$mathcal{C}$ we use the Koszul duality between the spectral Lie operad and the\u0000cocommutative cooperad to construct an enveloping Hopf algebra functor\u0000$mathcal{U}: mathrm{Alg}_{mathrm{Lie}}(mathcal{C}) to\u0000mathrm{Hopf}(mathcal{C})$ from spectral Lie algebras in $mathcal{C}$ to\u0000cocommutative Hopf algebras in $mathcal{C}$ left adjoint to a functor of\u0000derived primitive elements. We prove that if $mathcal{C}$ is a rational stable\u0000presentably symmetric monoidal $infty$-category, the enveloping Hopf algebra\u0000functor is fully faithful. We conclude that Lie algebras in $mathcal{C}$ are\u0000algebras over the monad underlying the adjunction $T simeq mathcal{U} circ\u0000mathrm{Lie}: mathcal{C} rightleftarrows\u0000mathrm{Alg}_{mathrm{Lie}}(mathcal{C}) to mathrm{Hopf}(mathcal{C}), $\u0000where $mathrm{Lie}$ is the free Lie algebra and $mathrm{T}$ is the tensor\u0000algebra. For general $mathcal{C}$ we introduce the notion of restricted\u0000$L_infty$-algebra as an algebra over the latter adjunction. For any field $K$\u0000we construct a forgetful functor from restricted Lie algebras in connective\u0000$H(K)$-modules to the $infty$-category underlying a right induced model\u0000structure on simplicial restricted Lie algebras over $K $.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195600","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}
Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee
In this paper we generalize the discrete r-homotopy to the discrete (s, r)-homotopy. Then by this notion, we introduce the discrete motion planning for robots which can move discreetly. Moreover, in this case the number of motion planning, called discrete topological complexity, required for these robots is reduced. Then we prove some properties of discrete topological complexity; For instance, we show that a discrete motion planning in a metric space X exists if and only if X is a discrete contractible space. Also, we prove that the discrete topological complexity depends only on the strictly discrete homotopy type of spaces.
在本文中，我们将离散 R 同调概括为离散 (s,r) 同调。然后，根据这一概念，我们为可以离散移动的机器人引入了离散运动规划。此外，在这种情况下，这些机器人所需的运动规划次数（称为离散拓扑复杂性）也会减少。然后，我们证明了离散拓扑复杂性的一些性质；例如，我们证明了如果且仅当 X 是离散可收缩空间时，才存在度量空间 X 中的离散运动规划。此外，我们还证明了离散拓扑复杂性只取决于空间的严格离散同调类型。
{"title":"A Discrete Topological Complexity of Discrete Motion Planning","authors":"Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee","doi":"arxiv-2408.05858","DOIUrl":"https://doi.org/arxiv-2408.05858","url":null,"abstract":"In this paper we generalize the discrete r-homotopy to the discrete (s,\u0000r)-homotopy. Then by this notion, we introduce the discrete motion planning for\u0000robots which can move discreetly. Moreover, in this case the number of motion\u0000planning, called discrete topological complexity, required for these robots is\u0000reduced. Then we prove some properties of discrete topological complexity; For\u0000instance, we show that a discrete motion planning in a metric space X exists if\u0000and only if X is a discrete contractible space. Also, we prove that the\u0000discrete topological complexity depends only on the strictly discrete homotopy\u0000type of spaces.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195601","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 complex variety $X$ admits a emph{cellular resolution of singularities} if there exists a resolution of singularities $widetilde Xto X$ such that its exceptional locus as well as $widetilde X$ and the singular locus of $X$ admit a cellular decomposition. We give a concrete description of the motive with compact support of $X$ in terms of its Borel--Moore homology, under some mild conditions. We give many examples, including rational projective curves and toric varieties of dimension two and three.
{"title":"The motive of a variety with cellular resolution of singularities","authors":"Bruno Stonek","doi":"arxiv-2408.05766","DOIUrl":"https://doi.org/arxiv-2408.05766","url":null,"abstract":"A complex variety $X$ admits a emph{cellular resolution of singularities} if\u0000there exists a resolution of singularities $widetilde Xto X$ such that its\u0000exceptional locus as well as $widetilde X$ and the singular locus of $X$ admit\u0000a cellular decomposition. We give a concrete description of the motive with\u0000compact support of $X$ in terms of its Borel--Moore homology, under some mild\u0000conditions. We give many examples, including rational projective curves and\u0000toric varieties of dimension two and three.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225159","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 the present work is a comparative study of different persistence kernels applied to various classification problems. After some necessary preliminaries on homology and persistence diagrams, we introduce five different kernels that are then used to compare their performances of classification on various datasets. We also provide the Python codes for the reproducibility of results.
{"title":"Persistence kernels for classification: A comparative study","authors":"Cinzia Bandiziol, Stefano De Marchi","doi":"arxiv-2408.07090","DOIUrl":"https://doi.org/arxiv-2408.07090","url":null,"abstract":"The aim of the present work is a comparative study of different persistence\u0000kernels applied to various classification problems. After some necessary\u0000preliminaries on homology and persistence diagrams, we introduce five different\u0000kernels that are then used to compare their performances of classification on\u0000various datasets. We also provide the Python codes for the reproducibility of\u0000results.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195602","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}
Quasi-elliptic cohomology is conjectured by Sati and Schreiber as a particularly suitable approximation to equivariant 4-th Cohomotopy, which classifies the charges carried by M-branes in M-theory in a way that is analogous to the traditional idea that complex K-theory classifies the charges of D-branes in string theory. In this paper we compute quasi-elliptic cohomology of 4-spheres under the action by some finite subgroups that are the most interesting isotropy groups where the M5-branes may sit.
准椭圆同调学（Quasi-elliptic cohomology）是萨提（Sati）和施雷伯（Schreiber）的猜想，它是等变 4-th 同调学（Equivariant 4-th Cohomotopy）的一个特别合适的近似，它将 M 理论中的 M 粒子所带的电荷进行了分类，这与复 K 理论将弦理论中的 D 粒子所带的电荷进行分类的传统观点类似。在本文中，我们计算了一些有限子群作用下 4 球的准椭圆全同调，这些有限子群是最有趣的等向群，M5-branes 可能就位于这些等向群中。
{"title":"Quasi-elliptic cohomology of 4-spheres","authors":"Zhen Huan","doi":"arxiv-2408.02278","DOIUrl":"https://doi.org/arxiv-2408.02278","url":null,"abstract":"Quasi-elliptic cohomology is conjectured by Sati and Schreiber as a\u0000particularly suitable approximation to equivariant 4-th Cohomotopy, which\u0000classifies the charges carried by M-branes in M-theory in a way that is\u0000analogous to the traditional idea that complex K-theory classifies the charges\u0000of D-branes in string theory. In this paper we compute quasi-elliptic\u0000cohomology of 4-spheres under the action by some finite subgroups that are the\u0000most interesting isotropy groups where the M5-branes may sit.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944807","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 $gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone $gamma$. So far, the computation of the $gamma$-linear projected barcode has only been studied in the functional setting, in which persistence modules come from the persistent cohomology of $mathbb{R}^n$-valued functions. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over $mathbb{R}^n$ that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of $mathbb{R}^n$ and a barcode template in each face of the arrangement; second, given any query linear form $u$ in the polar of $gamma$, it locates $u$ within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.
{"title":"Computation of $γ$-linear projected barcodes for multiparameter persistence","authors":"Alex Fernandes, Steve Oudot, Francois Petit","doi":"arxiv-2408.01065","DOIUrl":"https://doi.org/arxiv-2408.01065","url":null,"abstract":"The $gamma$-linear projected barcode was recently introduced as an\u0000alternative to the well-known fibered barcode for multiparameter persistence,\u0000in which restrictions of the modules to lines are replaced by pushforwards of\u0000the modules along linear forms in the polar of some fixed cone $gamma$. So\u0000far, the computation of the $gamma$-linear projected barcode has only been\u0000studied in the functional setting, in which persistence modules come from the\u0000persistent cohomology of $mathbb{R}^n$-valued functions. Here we develop a\u0000method that works in the algebraic setting directly, for any multiparameter\u0000persistence module over $mathbb{R}^n$ that is given via a finite free\u0000resolution. Our approach is similar to that of RIVET: first, it pre-processes\u0000the resolution to build an arrangement in the dual of $mathbb{R}^n$ and a\u0000barcode template in each face of the arrangement; second, given any query\u0000linear form $u$ in the polar of $gamma$, it locates $u$ within the arrangement\u0000to produce the corresponding barcode efficiently. While our theoretical\u0000complexity bounds are similar to the ones of RIVET, our arrangement turns out\u0000to be simpler thanks to the linear structure of the space of linear forms. Our\u0000theoretical analysis combines sheaf-theoretic and module-theoretic techniques,\u0000showing that multiparameter persistence modules can be converted into a special\u0000type of complexes of sheaves on vector spaces called conic-complexes, whose\u0000derived pushforwards by linear forms have predictable barcodes.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944857","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 $mathbb{F}_2$-synthetic Adams spectral sequence. We obtain new computational information about $mathbb{C}$-motivic and classical stable homotopy groups.
{"title":"Classical stable homotopy groups of spheres via $mathbb{F}_2$-synthetic methods","authors":"Robert Burklund, Daniel C. Isaksen, Zhouli Xu","doi":"arxiv-2408.00987","DOIUrl":"https://doi.org/arxiv-2408.00987","url":null,"abstract":"We study the $mathbb{F}_2$-synthetic Adams spectral sequence. We obtain new\u0000computational information about $mathbb{C}$-motivic and classical stable\u0000homotopy groups.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944858","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 investigate the facets of the Vietoris--Rips complex $mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $mathcal{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.
{"title":"Facets in the Vietoris--Rips complexes of hypercubes","authors":"Joseph Briggs, Ziqin Feng, Chris Wells","doi":"arxiv-2408.01288","DOIUrl":"https://doi.org/arxiv-2408.01288","url":null,"abstract":"In this paper, we investigate the facets of the Vietoris--Rips complex\u0000$mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We\u0000are particularly interested in those facets which are somehow independent of\u0000the dimension $n$. Using Hadamard matrices, we prove that the number of\u0000different dimensions of such facets is a super-polynomial function of the scale\u0000$r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th\u0000dimensional homology of the complex $mathcal{VR}(Q_n; r)$ is non-trivial when\u0000$n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944809","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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