arXiv - MATH - Algebraic Topology最新文献_第4页

On the mod-2 cohomology of the product of the infinite lens space and the space of invariants in a generic degree 论无限透镜空间与一般度不变空间乘积的模-2同调

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-14 DOI: arxiv-2408.07485

Dang Vo Phuc

{"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}

引用次数: 0

A counter-example to Singer's conjecture for the algebraic transfer 辛格代数转移猜想的反例

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-13 DOI: arxiv-2408.06669

Nguyen Sum

{"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}

引用次数: 0

Restricted $L_infty$-algebras and a derived Milnor-Moore theorem 受限$L_infty$-代数和衍生米尔诺-摩尔定理

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-13 DOI: arxiv-2408.06917

Hadrian Heine

{"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}

引用次数: 0

A Discrete Topological Complexity of Discrete Motion Planning 离散运动规划的离散拓扑复杂性

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-11 DOI: arxiv-2408.05858

Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee

引用次数: 0

The motive of a variety with cellular resolution of singularities 具有奇点细胞解法的品种动机

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-11 DOI: arxiv-2408.05766

Bruno Stonek

引用次数: 0

Persistence kernels for classification: A comparative study 用于分类的持久性内核：比较研究

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-09 DOI: arxiv-2408.07090

Cinzia Bandiziol, Stefano De Marchi

引用次数: 0

Quasi-elliptic cohomology of 4-spheres 4 球体的准椭圆同调

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-05 DOI: arxiv-2408.02278

Zhen Huan

引用次数: 0

Computation of $γ$-linear projected barcodes for multiparameter persistence 计算多参数持久性的 $γ$ 线性投影条形码

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-02 DOI: arxiv-2408.01065

Alex Fernandes, Steve Oudot, Francois Petit

{"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}

引用次数: 0

Classical stable homotopy groups of spheres via $mathbb{F}_2$-synthetic methods 通过$mathbb{F}_2$合成方法研究球面的经典稳定同调群

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-02 DOI: arxiv-2408.00987

Robert Burklund, Daniel C. Isaksen, Zhouli Xu

引用次数: 0

Facets in the Vietoris--Rips complexes of hypercubes 超立方体的Vietoris--Rips复合体中的刻面

arXiv - MATH - Algebraic Topology Pub Date : 2024-08-02 DOI: arxiv-2408.01288

Joseph Briggs, Ziqin Feng, Chris Wells

引用次数: 0