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The low dimensional homology groups of the elementary group of degree two 二阶初等群的低维同调群
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-03-04 DOI: 10.1007/s40062-026-00401-3
Behrooz Mirzaii, Elvis Torres Pérez
{"title":"The low dimensional homology groups of the elementary group of degree two","authors":"Behrooz Mirzaii,&nbsp;Elvis Torres Pérez","doi":"10.1007/s40062-026-00401-3","DOIUrl":"10.1007/s40062-026-00401-3","url":null,"abstract":"<div><p>In this article, we study the first, second, and third homology groups of the elementary group <span>({textrm{E}}_2(A))</span>, where <i>A</i> is a commutative ring. In particular, we establish a refined Bloch–Wigner type exact sequence over a semilocal ring (subject to mild restrictions on its residue fields) in which either <span>(-1 in ({A^times })^2)</span> or <span>(|{A^times }/({A^times })^2| le 4)</span>.\u0000</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 3","pages":"509 - 535"},"PeriodicalIF":0.5,"publicationDate":"2026-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148613022","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
Relative cohomological dimension of a relatively hyperbolic pair 相对双曲对的相对上同调维数
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-02-26 DOI: 10.1007/s40062-026-00403-1
Harsh Patil
{"title":"Relative cohomological dimension of a relatively hyperbolic pair","authors":"Harsh Patil","doi":"10.1007/s40062-026-00403-1","DOIUrl":"10.1007/s40062-026-00403-1","url":null,"abstract":"<div><p>We show that the relative cohomological dimension <span>(textrm{cd}_R(G,H))</span> of a relatively hyperbolic pair (<i>G</i>, <i>H</i>) is always finite when <i>G</i> does not contain <i>R</i>-torsion. We also show that this dimension is preserved under quasi-isometries, provided that <i>G</i> is torsion-free and the peripheral subgroup <i>H</i> is unconstricted and of type <span>(F_{infty })</span>. As a corollary of our methods, we compute <span>(textrm{cd}_{mathbb {Z}}(G,H))</span> in several cases.\u0000</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 3","pages":"491 - 508"},"PeriodicalIF":0.5,"publicationDate":"2026-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-026-00403-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On (textrm{K}_1)-stability of the relative DSER elementary orthogonal group 相对DSER初等正交群的(textrm{K}_1) -稳定性
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-02-24 DOI: 10.1007/s40062-026-00402-2
A. A. Ambily, Gayathry Pradeep
{"title":"On (textrm{K}_1)-stability of the relative DSER elementary orthogonal group","authors":"A. A. Ambily,&nbsp;Gayathry Pradeep","doi":"10.1007/s40062-026-00402-2","DOIUrl":"10.1007/s40062-026-00402-2","url":null,"abstract":"<div><p>In this paper, we study the stabilization map on the relative <i>Dickson-Siegel-Eichler-Roy</i> (DSER) elementary orthogonal group. We obtain two bounds for the hyperbolic rank of the underlying quadratic space such that the stabilization map on <span>(textrm{K}_1)</span>- analogue of the relative <i>DSER</i> elementary orthogonal group becomes an isomorphism and a monomorphism.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 3","pages":"481 - 490"},"PeriodicalIF":0.5,"publicationDate":"2026-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148613833","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
Commutative classifying space for simplicial groups 简单群的交换分类空间
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-02-19 DOI: 10.1007/s40062-026-00397-w
Cihan Okay, Pál Zsámboki
{"title":"Commutative classifying space for simplicial groups","authors":"Cihan Okay,&nbsp;Pál Zsámboki","doi":"10.1007/s40062-026-00397-w","DOIUrl":"10.1007/s40062-026-00397-w","url":null,"abstract":"<div><p>In this paper, we introduce a simplicial analog of classifying spaces for commutativity which classify principal bundles with commutativity structure on their transition functions. Our construction <span>(overline{W}(tau ,K))</span>, which takes as input a simplicial group <i>K</i> and a cosimplicial group <span>(tau )</span> that encodes the additional structure such as commutativity, is a variation of the <span>(overline{W})</span>-construction for simplicial groups. Our main result shows that the geometric realization of our <span>(overline{W}(tau ,K))</span> is homotopy equivalent to the topological classifying space <span>(B(tau ,|K|))</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 3","pages":"455 - 480"},"PeriodicalIF":0.5,"publicationDate":"2026-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-026-00397-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148613548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rigid models for 2-gerbes I: Chern–Simons geometry 2-gerbes的刚性模型I: chen - simons几何
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-02-18 DOI: 10.1007/s40062-025-00381-w
David Michael Roberts, Raymond F. Vozzo
{"title":"Rigid models for 2-gerbes I: Chern–Simons geometry","authors":"David Michael Roberts,&nbsp;Raymond F. Vozzo","doi":"10.1007/s40062-025-00381-w","DOIUrl":"10.1007/s40062-025-00381-w","url":null,"abstract":"<div><p>Motivated by the problem of constructing explicit geometric string structures, we give a rigid model for bundle 2-gerbes, and define connective structures thereon. This model is designed to make explicit calculations easier in applications to physics. To compare to the existing definition, we give a functorial construction of a bundle 2-gerbe as in the literature from our rigid model, including with connections. As an example we prove that the Chern–Simons bundle 2-gerbe from the literature, with its connective structure, can be rigidified—it arises, up to <i>isomorphism</i> in the strongest possible sense, from a rigid bundle 2-gerbe with connective structure via this construction. Further, our rigid version of 2-gerbe trivialisation (with connections) gives rise to trivialisations (with connections) of bundle 2-gerbes in the usual sense, and as such can be used to describe geometric string structures. The preprint of this article is available as arXiv:2209.05521.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 3","pages":"389 - 454"},"PeriodicalIF":0.5,"publicationDate":"2026-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148613251","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 stability of persistent homology of hypergraphs 超图的持久同调的稳定性
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-02-13 DOI: 10.1007/s40062-026-00400-4
Shiquan Ren, Jie Wu
{"title":"The stability of persistent homology of hypergraphs","authors":"Shiquan Ren,&nbsp;Jie Wu","doi":"10.1007/s40062-026-00400-4","DOIUrl":"10.1007/s40062-026-00400-4","url":null,"abstract":"<div><p>Hypergraphs are the most general model for complex networks involving group interactions. Taking the ideas of path homology from Grigor’yan et al. ((Homologies of path complexes and digraphs. arXiv:1207.2834, 2013; Torsion of digraphs and path complexes. arXiv:2012.07302v1, 2020; Pure Appl Math Q 10(4):619–674 2014; Asian J Math 15(5):887–932, 2015; J Math Sci 248(5):564–599, 2020), Bressan et al. (Asian J Math 23(3):479–500, 2019) introduced embedded homology of hypergraphs, which has leaded to successful applications in protein-ligand binding network (Liu et al. in Brief Bioinform. https://doi.org/10.1093/bib/bbaa411, 2021; Brief Bioinform 22(5):bbab127, 2021). A fundamental question arising from practical applications is about the stability of the persistent embedded homology of hypergraphs. In this paper, we prove the stability of the persistent embedded homology as well as the persistent homology of the associated simplicial complex with respect to perturbations of the filtration on a hypergraph. We apply the persistent homology methods to morphisms of hypergraphs and prove the stability with respect to perturbations of the filtrations. We prove the constancy of the persistent Betti numbers under some conditions on the simple-homotopy types of hypergraphs.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 3","pages":"347 - 387"},"PeriodicalIF":0.5,"publicationDate":"2026-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148613519","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
Graphs arising from the dual Steenrod algebra 由对偶Steenrod代数产生的图
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-02-03 DOI: 10.1007/s40062-026-00394-z
Connor Elliott, Courtney Hauf, Kai Morton, Sarah Petersen, Leticia Schow
{"title":"Graphs arising from the dual Steenrod algebra","authors":"Connor Elliott,&nbsp;Courtney Hauf,&nbsp;Kai Morton,&nbsp;Sarah Petersen,&nbsp;Leticia Schow","doi":"10.1007/s40062-026-00394-z","DOIUrl":"10.1007/s40062-026-00394-z","url":null,"abstract":"<div><p>We extend Wood’s graph theoretic interpretation of certain quotients of the mod 2 dual Steenrod algebra to quotients of the mod <i>p</i> dual Steenrod algebra where <i>p</i> is an odd prime and to quotients of the <span>(C_2)</span>-equivariant dual Steenrod algebra. We establish connectedness criteria for graphs associated to monomials in these algebra quotients and investigate questions about trees and Hamilton cycles in these settings. We also give graph theoretic interpretations of algebraic structures such as the coproduct and antipode arising from the Hopf algebra structure on the mod <i>p</i> dual Steenrod algebra and the Hopf algebroid structure of the <span>(C_2)</span>-equivariant dual Steenrod algebra.\u0000</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 2","pages":"307 - 345"},"PeriodicalIF":0.5,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-026-00394-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147727341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Configuration spaces of circles in the plane 平面上圆的位形空间
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-01-27 DOI: 10.1007/s40062-026-00398-9
Justin M. Curry, Ryan C. Gelnett, Matthew C. B. Zaremsky
{"title":"Configuration spaces of circles in the plane","authors":"Justin M. Curry,&nbsp;Ryan C. Gelnett,&nbsp;Matthew C. B. Zaremsky","doi":"10.1007/s40062-026-00398-9","DOIUrl":"10.1007/s40062-026-00398-9","url":null,"abstract":"<div><p>We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these fundamental groups are obtained as iterated semidirect products of subgroups of braid groups, with the structure for each component dictated by a finite rooted tree. These groups can be viewed as “braided” versions of the automorphism groups of such trees. We also discuss connections to statistical mechanics, topological data analysis, and geometric group theory.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 2","pages":"277 - 305"},"PeriodicalIF":0.5,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147727623","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
Unstable arithmetic fracture squares in (infty )-topoi (infty ) -拓扑中的不稳定算术断裂平方
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-01-23 DOI: 10.1007/s40062-026-00396-x
Klaus Mattis
{"title":"Unstable arithmetic fracture squares in (infty )-topoi","authors":"Klaus Mattis","doi":"10.1007/s40062-026-00396-x","DOIUrl":"10.1007/s40062-026-00396-x","url":null,"abstract":"<div><p>We show that for a large class of <span>(infty )</span>-topoi there exist unstable arithmetic fracture squares, i.e. squares which recover a nilpotent sheaf <i>F</i>as the pullback of the rationalization of <i>F</i> with the product of the <i>p</i>-completions of <i>F</i> ranging over all primes <span>(pin {mathbb {Z}})</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 2","pages":"245 - 276"},"PeriodicalIF":0.5,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-026-00396-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147727654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
(textrm{THH}) of the Morava E-theory spectrum (E_2) (textrm{THH}) 莫拉瓦e理论谱 (E_2)
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2026-01-14 DOI: 10.1007/s40062-025-00393-6
Sanjana Agarwal
{"title":"(textrm{THH}) of the Morava E-theory spectrum (E_2)","authors":"Sanjana Agarwal","doi":"10.1007/s40062-025-00393-6","DOIUrl":"10.1007/s40062-025-00393-6","url":null,"abstract":"<div><p>The Morava <i>E</i>-theories, <span>(E_{n})</span>, are complex-oriented 2-periodic ring spectra, with homotopy groups <span>(W_{{{mathbb {F}}}_{p^{n}}}[[u_{1}, u_{2},ldots , u_{n-1}]][u,u^{-1}])</span>. Here <i>W</i> denotes the ring of Witt vectors. <span>(E_{n})</span> is a Landweber exact spectrum and hence uniquely determined by its homotopy groups as <span>(BP_{*})</span>-algebra. Algebraic <i>K</i>-theory of <span>(E_{n})</span> is a key ingredient towards analyzing the layers in the <i>p</i>-complete Waldhausen’s algebraic <i>K</i>-theory chromatic tower. One hopes to use the machinery of trace methods to get results towards algebraic <i>K</i>-theory once the computation for <span>(THH(E_{n}))</span> is known. In this paper we describe <span>(THH(E_{2}))</span> as part of consecutive chain of cofiber sequences where each cofiber sits in the next cofiber sequence and the first term of each cofiber sequence is describable completely in terms of suspensions and localizations of <span>(E_{2})</span>. For these results, we first calculate <i>K</i>(<i>i</i>)-homology of <span>(THH(E_{2}))</span> using a Bökstedt spectral sequence and then lift the generating classes of <i>K</i>(1)-homology to fundamental classes in homotopy group of <span>(THH(E_{2}))</span>. These lifts allow us to construct terms of the cofiber sequence and explicitly understand how they map to <span>(THH(E_{2}))</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 2","pages":"211 - 244"},"PeriodicalIF":0.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00393-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147727396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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