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Homology Homotopy and Applications最新文献

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Sharpness of saturated fusion systems on a Sylow $p$-subgroup of $mathrm{G}_2 (p)$ 在$ mathm {G}_2 (p)$的Sylow $p$-子群上饱和聚变系统的锐性
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-22 DOI: 10.4310/hha.2023.v25.n2.a14
Valentina Grazian, Ettore Marmo
{"title":"Sharpness of saturated fusion systems on a Sylow $p$-subgroup of $mathrm{G}_2 (p)$","authors":"Valentina Grazian, Ettore Marmo","doi":"10.4310/hha.2023.v25.n2.a14","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a14","url":null,"abstract":"We prove that the Díaz–Park sharpness conjecture holds for saturated fusion systems defined on a Sylow $p$-subgroup of the group $mathrm{G}_2 (p)$, for $p geqslant 5$.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Classifying space via homotopy coherent nerve 利用同伦相干神经对空间进行分类
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-22 DOI: 10.4310/hha.2023.v25.n2.a16
Kensuke Arakawa
{"title":"Classifying space via homotopy coherent nerve","authors":"Kensuke Arakawa","doi":"10.4310/hha.2023.v25.n2.a16","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a16","url":null,"abstract":"We prove that the classifying space of a simplicial group is modeled by its homotopy coherent nerve. We will also show that the claim remains valid for simplicial groupoids.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
$K$-theory of real Grassmann manifolds 真实格拉斯曼流形的K理论
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-22 DOI: 10.4310/hha.2023.v25.n2.a17
Sudeep Podder, Parameswaran Sankaran
{"title":"$K$-theory of real Grassmann manifolds","authors":"Sudeep Podder, Parameswaran Sankaran","doi":"10.4310/hha.2023.v25.n2.a17","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a17","url":null,"abstract":"Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $mathbb{R}^n$. We compute the complex $K$-ring of $G_{n,k}:$, up to a small indeterminacy, for all values of $n,k$ where $2 leqslant k leqslant n - 2$. When $n equiv 0 (operatorname{mod} 4), k equiv 1 (operatorname{mod} 2)$, we use the Hodgkin spectral sequence to determine the $K$-ring completely.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On bialgebras, comodules, descent data and Thom spectra in $infty$-categories 论$infty$ -范畴中的双代数、模、下降数据和Thom谱
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-01 DOI: 10.4310/hha.2023.v25.n2.a10
Jonathan Beardsley
{"title":"On bialgebras, comodules, descent data and Thom spectra in $infty$-categories","authors":"Jonathan Beardsley","doi":"10.4310/hha.2023.v25.n2.a10","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a10","url":null,"abstract":"This paper establishes several results for coalgebraic structure in $infty$-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. We prove that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category (as opposed to a relative (co)tensor product over the underlying algebra or coalgebra of the bialgebra). We give two examples of higher coalgebraic structure: first, following Hess we show that for a map of $mathbb{E}_n$-ring spectra $varphi : A to B$, the associated $infty$-category of descent data is equivalent to the $infty$-category of comodules over $B otimes_A B$, the so-called descent coring; secondly, we show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the $infty$-categorical Thom diagonal of Ando, Blumberg, Gepner, Hopkins and Rezk (which we describe explicitly) and that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way indicating that Thom spectra are good examples of spectral noncommutative torsors.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Zig-zag modules: cosheaves and $k$-theory z形模块:cosheaves和$k$-理论
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-11-01 DOI: 10.4310/hha.2023.v25.n2.a11
Ryan Grady, Anna Schenfisch
{"title":"Zig-zag modules: cosheaves and $k$-theory","authors":"Ryan Grady, Anna Schenfisch","doi":"10.4310/hha.2023.v25.n2.a11","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a11","url":null,"abstract":"Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig‑zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig‑zag modules and the combinatorial entrance path category on stratified $mathbb{R}$. Finally, we compute the algebraic $K$-theory of generalized zig‑zag modules and describe connections to both Euler curves and $K_0$ of the monoid of persistence diagrams as described by Bubenik and Elchesen.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Homology transfer products on free loop spaces: orientation reversal on spheres 自由环空间上的同调转移积:球面上的取向反转
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-10-11 DOI: 10.4310/hha.2023.v25.n2.a7
Philippe Kupper
{"title":"Homology transfer products on free loop spaces: orientation reversal on spheres","authors":"Philippe Kupper","doi":"10.4310/hha.2023.v25.n2.a7","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a7","url":null,"abstract":"We consider the space $Lambda M := H^1 (S^1, M)$ of loops of Sobolev class $H^1$ of a compact smooth manifold $M$, the so-called free loop space of $M$. We take quotients $Lambda M / G$ where $G$ is a finite subgroup of $O(2)$ acting by linear reparametrization of $S^1$. We use the existence of transfer maps $operatorname{tr} : H_ast (Lambda M / G) to H_ast (Lambda M)$ to define a homology product on $Lambda M / G$ via the Chas–Sullivan loop product. We call this product $P_G$ the transfer product. The involution $vartheta : Lambda M to Lambda M$ which reverses orientation, $vartheta ( gamma (t) := gamma (1-t)$, is of particular interest to us. We compute $H_ast (Lambda S^n / vartheta ; mathbb{Q}), n gt 2$, and the product[P_vartheta : H_i (Lambda S^n / vartheta ; mathbb{Q}) times H_j (Lambda S^n / vartheta ; mathbb{Q)} to H_{i+j-n} (Lambda Sn/vartheta ; mathbb{Q})]associated to orientation reversal. Rationally Pvartheta can be realized “geometrically” using the concatenation of equivalence classes of loops. There is a qualitative difference between the homology of $Lambda S^n / vartheta$ and the homology of $Lambda S^n / G$ when $G subset S^1 subset O(2)$ does not “contain” the orientation reversal. This might be interesting with respect to possible differences in the number of closed geodesics between non-reversible and reversible Finsler metrics on $S^n$, the latter might always be infinite.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Haefliger’s approach for spherical knots modulo immersions 球节模浸的Haefliger方法
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-10-04 DOI: 10.4310/hha.2023.v25.n2.a4
Neeti Gauniyal
{"title":"Haefliger’s approach for spherical knots modulo immersions","authors":"Neeti Gauniyal","doi":"10.4310/hha.2023.v25.n2.a4","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n2.a4","url":null,"abstract":"$defEmb{overline{Emb}}$We show that for the spaces of spherical embeddings modulo immersions $Emb (S^n, S^{n+q})$ and long embeddings modulo immersions $Emb_partial (D^n, D^{n+q})$, the set of connected components is isomorphic to $pi_{n+1} (SG, SG_q)$ for $q geqslant 3$. As a consequence, we show that all the terms of the long exact sequence of the triad $(SG; SO, SG_q)$ have a geometric meaning relating to spherical embeddings and immersions.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homotopy types of truncated projective resolutions 截断投影分辨率的同调类型
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-08-23 DOI: 10.4310/HHA.2007.v9.n2.a16
W. Mannan
{"title":"Homotopy types of truncated projective resolutions","authors":"W. Mannan","doi":"10.4310/HHA.2007.v9.n2.a16","DOIUrl":"https://doi.org/10.4310/HHA.2007.v9.n2.a16","url":null,"abstract":"We work over an arbitrary ring R. Given two truncated projective\u0000resolutions of equal length for the same module, we consider\u0000their underlying chain complexes. We show they may be\u0000stabilized by projective modules to obtain a pair of complexes\u0000of the same homotopy type","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42199405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Duality in the homology of 5-manifolds 5流形同调中的对偶性
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-08-23 DOI: 10.4310/HHA.2017.v19.n1.a9
W. Mannan
{"title":"Duality in the homology of 5-manifolds","authors":"W. Mannan","doi":"10.4310/HHA.2017.v19.n1.a9","DOIUrl":"https://doi.org/10.4310/HHA.2017.v19.n1.a9","url":null,"abstract":"We show that the homological properties of a 5-manifold M with fundamental group G are encapsulated in a G-invariant stable form on the dual of the third syzygy of Z. In this notation one may express an even stronger version of Poincare duality for M. However we find an obstruction to this duality.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41756829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The homotopy types of $Sp(n)$-gauge groups over $mathbb{C}P^2$ $mathbb{C}P^2$上$Sp(n)$-规范群的同伦类型
IF 0.5 4区 数学
Homology Homotopy and Applications Pub Date : 2023-01-01 DOI: 10.4310/hha.2023.v25.n1.a11
Sajjad Mohammadi
{"title":"The homotopy types of $Sp(n)$-gauge groups over $mathbb{C}P^2$","authors":"Sajjad Mohammadi","doi":"10.4310/hha.2023.v25.n1.a11","DOIUrl":"https://doi.org/10.4310/hha.2023.v25.n1.a11","url":null,"abstract":"","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70435079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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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