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Diffeological principal bundles and principal infinity bundles 差分主束和主无穷束
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-04-15 DOI: 10.1007/s40062-024-00347-4
Emilio Minichiello
{"title":"Diffeological principal bundles and principal infinity bundles","authors":"Emilio Minichiello","doi":"10.1007/s40062-024-00347-4","DOIUrl":"10.1007/s40062-024-00347-4","url":null,"abstract":"<div><p>In this paper, we study diffeological spaces as certain kinds of discrete simplicial presheaves on the site of cartesian spaces with the coverage of good open covers. The Čech model structure on simplicial presheaves provides us with a notion of <span>(infty )</span>-stack cohomology of a diffeological space with values in a diffeological abelian group <i>A</i>. We compare <span>(infty )</span>-stack cohomology of diffeological spaces with two existing notions of Čech cohomology for diffeological spaces in the literature Krepski et al. (Sheaves, principal bundles, and Čech cohomology for diffeological spaces. (2021). arxiv:2111 01032 [math.DG]), Iglesias-Zemmour (Čech-de-Rham Bicomplex in Diffeology (2020). http://math.huji.ac.il/piz/documents/CDRBCID.pdf). Finally, we prove that for a diffeological group <i>G</i>, that the nerve of the category of diffeological principal <i>G</i>-bundles is weak homotopy equivalent to the nerve of the category of <i>G</i>-principal <span>(infty )</span>-bundles on <i>X</i>, bridging the bundle theory of diffeology and higher topos theory.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"181 - 237"},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560997","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
A Matsumoto type theorem for (GL_n) over rings of non-commutative Laurent polynomials 非交换月桂多项式环上 $$GL_n$$ 的松本类型定理
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-04-06 DOI: 10.1007/s40062-024-00345-6
Ryusuke Sugawara
{"title":"A Matsumoto type theorem for (GL_n) over rings of non-commutative Laurent polynomials","authors":"Ryusuke Sugawara","doi":"10.1007/s40062-024-00345-6","DOIUrl":"10.1007/s40062-024-00345-6","url":null,"abstract":"<div><p>We give a Matsumoto-type presentation of <span>(K_2)</span>-groups over rings of non-commutative Laurent polynomials, which is a non-commutative version of M. Tomie’s result for loop groups. Our main idea is induced by U. Rehmann’s approach in the case of division rings.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"151 - 180"},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561000","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
A reasonable notion of dimension for singular intersection homology 奇异交点同调的合理维度概念
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-04-04 DOI: 10.1007/s40062-024-00343-8
David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré
{"title":"A reasonable notion of dimension for singular intersection homology","authors":"David Chataur,&nbsp;Martintxo Saralegi-Aranguren,&nbsp;Daniel Tanré","doi":"10.1007/s40062-024-00343-8","DOIUrl":"10.1007/s40062-024-00343-8","url":null,"abstract":"<div><p>M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces <i>S</i> of an Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing <i>S</i>. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing <i>S</i>. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann’s CS sets. In terms of King’s paper, this means that polyhedral dimension is a “reasonable” dimension. The proof uses a Mayer-Vietoris argument which needs an adapted subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"121 - 150"},"PeriodicalIF":0.7,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560999","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
Adams operations on the twisted K-theory of compact Lie groups 紧凑李群扭曲 K 理论上的亚当斯运算
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-03-18 DOI: 10.1007/s40062-024-00342-9
Chi-Kwong Fok
{"title":"Adams operations on the twisted K-theory of compact Lie groups","authors":"Chi-Kwong Fok","doi":"10.1007/s40062-024-00342-9","DOIUrl":"10.1007/s40062-024-00342-9","url":null,"abstract":"<div><p>In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted <i>K</i>-theory of connected, simply-connected and simple compact Lie groups <i>G</i>, in both equivariant and nonequivariant settings.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"99 - 120"},"PeriodicalIF":0.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149962","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 Vietoris–Rips complexes of finite metric spaces with scale 2 论尺度为 2 的有限度量空间的 Vietoris-Rips 复数
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-02-03 DOI: 10.1007/s40062-024-00340-x
Ziqin Feng, Naga Chandra Padmini Nukala
{"title":"On Vietoris–Rips complexes of finite metric spaces with scale 2","authors":"Ziqin Feng,&nbsp;Naga Chandra Padmini Nukala","doi":"10.1007/s40062-024-00340-x","DOIUrl":"10.1007/s40062-024-00340-x","url":null,"abstract":"<div><p>We examine the homotopy types of Vietoris–Rips complexes on certain finite metric spaces at scale 2. We consider the collections of subsets of <span>([m]={1, 2, ldots , m})</span> equipped with symmetric difference metric <i>d</i>, specifically, <span>({mathcal {F}}^m_n)</span>, <span>({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+1})</span>, <span>({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+2})</span>, and <span>({mathcal {F}}_{preceq A}^m)</span>. Here <span>({mathcal {F}}^m_n)</span> is the collection of size <i>n</i> subsets of [<i>m</i>] and <span>({mathcal {F}}_{preceq A}^m)</span> is the collection of subsets <span>(preceq A)</span> where <span>(preceq )</span> is a total order on the collections of subsets of [<i>m</i>] and <span>(Asubseteq [m])</span> (see the definition of <span>(preceq )</span> in Sect. 1). We prove that the Vietoris–Rips complexes <span>({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}^m_n, 2))</span> and <span>({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+1}, 2))</span> are either contractible or homotopy equivalent to a wedge sum of <span>(S^2)</span>’s; also, the complexes <span>({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}_n^mcup {mathcal {F}}^m_{n+2}, 2))</span> and <span>({{mathcal {V}}}{{mathcal {R}}}({mathcal {F}}_{preceq A}^m, 2))</span> are either contractible or homotopy equivalent to a wedge sum of <span>(S^3)</span>’s. We provide inductive formulae for these homotopy types extending the result of Barmak about the independence complexes of Kneser graphs KG<span>(_{2, k})</span> and the result of Adamaszek and Adams about Vietoris–Rips complexes of hypercube graphs with scale 2.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 1","pages":"79 - 98"},"PeriodicalIF":0.7,"publicationDate":"2024-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139677567","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
Associative 2-algebras and nonabelian extensions of associative algebras 关联二元数和关联数的非阿贝尔扩展
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-02-01 DOI: 10.1007/s40062-024-00341-w
Yunhe Sheng, You Wang
{"title":"Associative 2-algebras and nonabelian extensions of associative algebras","authors":"Yunhe Sheng,&nbsp;You Wang","doi":"10.1007/s40062-024-00341-w","DOIUrl":"10.1007/s40062-024-00341-w","url":null,"abstract":"<div><p>In this paper, we study nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. First we construct an associative 2-algebra using the bimultipliers of an associative algebra. Then we classify nonabelian extensions of associative algebras using associative 2-algebra homomorphisms. Finally we analyze the relation between nonabelian extensions of associative algebras and nonabelian extensions of the corresponding commutator Lie algebras.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 1","pages":"63 - 77"},"PeriodicalIF":0.7,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139666617","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
Lambda module structure on higher K-groups 高 K 群上的 Lambda 模块结构
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-01-25 DOI: 10.1007/s40062-024-00339-4
Sourayan Banerjee, Vivek Sadhu
{"title":"Lambda module structure on higher K-groups","authors":"Sourayan Banerjee,&nbsp;Vivek Sadhu","doi":"10.1007/s40062-024-00339-4","DOIUrl":"10.1007/s40062-024-00339-4","url":null,"abstract":"<div><p>In this article, we show that for a quasicompact scheme <i>X</i> and <span>(n&gt;0,)</span> the <i>n</i>-th <i>K</i>-group <span>(K_{n}(X))</span> is a <span>(lambda )</span>-module over a <span>(lambda )</span>-ring <span>(K_{0}(X))</span> in the sense of Hesselholt.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 1","pages":"53 - 61"},"PeriodicalIF":0.7,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139561241","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
LHS-spectral sequences for regular extensions of categories 类的正则扩展的 LHS-谱序列
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-01-20 DOI: 10.1007/s40062-024-00338-5
Ergün Yalçın
{"title":"LHS-spectral sequences for regular extensions of categories","authors":"Ergün Yalçın","doi":"10.1007/s40062-024-00338-5","DOIUrl":"10.1007/s40062-024-00338-5","url":null,"abstract":"<div><p>In (Xu, J Pure Appl Algebra 212:2555–2569, 2008), a LHS-spectral sequence for target regular extensions of small categories is constructed. We extend this construction to ext-groups and construct a similar spectral sequence for source regular extensions (with right module coefficients). As a special case of these LHS-spectral sequences, we obtain three different versions of Słomińska’s spectral sequence for the cohomology of regular EI-categories. We show that many well-known spectral sequences related to the homology decompositions of finite groups, centric linking systems, and the orbit category of fusion systems can be obtained as the LHS-spectral sequence of an extension.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 1","pages":"1 - 51"},"PeriodicalIF":0.7,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139509147","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
Periodic self maps and thick ideals in the stable motivic homotopy category over ({mathbb {C}}) at odd primes ({mathbb {C}})上奇素数下稳定动机同伦范畴的周期自映射与厚理想
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-11-23 DOI: 10.1007/s40062-023-00337-y
Sven-Torben Stahn
{"title":"Periodic self maps and thick ideals in the stable motivic homotopy category over ({mathbb {C}}) at odd primes","authors":"Sven-Torben Stahn","doi":"10.1007/s40062-023-00337-y","DOIUrl":"10.1007/s40062-023-00337-y","url":null,"abstract":"<div><p>In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over <span>({mathbb {C}})</span>. In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 4","pages":"563 - 604"},"PeriodicalIF":0.5,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138473083","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 of the (KU_G)-local equivariant sphere spectrum (KU_G) -局部等变球谱的同伦
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2023-11-20 DOI: 10.1007/s40062-023-00336-z
Tanner N. Carawan, Rebecca Field, Bertrand J. Guillou, David Mehrle, Nathaniel J. Stapleton
{"title":"The homotopy of the (KU_G)-local equivariant sphere spectrum","authors":"Tanner N. Carawan,&nbsp;Rebecca Field,&nbsp;Bertrand J. Guillou,&nbsp;David Mehrle,&nbsp;Nathaniel J. Stapleton","doi":"10.1007/s40062-023-00336-z","DOIUrl":"10.1007/s40062-023-00336-z","url":null,"abstract":"<div><p>We compute the homotopy Mackey functors of the <span>(KU_G)</span>-local equivariant sphere spectrum when <i>G</i> is a finite <i>q</i>-group for an odd prime <i>q</i>, building on the degree zero case due to Bonventre and the third and fifth authors.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"18 4","pages":"543 - 561"},"PeriodicalIF":0.5,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138473129","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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