{"title":"Shellability of 3-cut complexes of squared cycle graphs","authors":"Pratiksha Chauhan, Samir Shukla, Kumar Vinayak","doi":"10.1007/s40062-025-00365-w","DOIUrl":"10.1007/s40062-025-00365-w","url":null,"abstract":"<div><p>For a positive integer <i>k</i>, the <i>k</i>-cut complex of a graph <i>G</i> is the simplicial complex whose facets are the <span>((|V(G)|-k))</span>-subsets <span>(sigma )</span> of the vertex set <i>V</i>(<i>G</i>) of <i>G</i> such that the induced subgraph of <i>G</i> on <span>(V(G) setminus sigma )</span> is disconnected. These complexes first appeared in the master thesis of Denker and were further studied by Bayer et al. (SIAM J Discrete Math 38(2):1630–1675, 2024). In the same article, Bayer et al. conjectured that for <span>(k ge 3)</span>, the <i>k</i>-cut complexes of squared cycle graphs are shellable. Moreover, they also conjectured about the Betti numbers of these complexes when <span>(k=3)</span>. In this article, we prove these conjectures for <span>(k=3)</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"163 - 193"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455437","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}
{"title":"On duoidal (infty )-categories","authors":"Takeshi Torii","doi":"10.1007/s40062-025-00364-x","DOIUrl":"10.1007/s40062-025-00364-x","url":null,"abstract":"<div><p>A duoidal category is a category equipped with two monoidal structures in which one is (op)lax monoidal with respect to the other. In this paper we introduce duoidal <span>(infty )</span>-categories which are counterparts of duoidal categories in the setting of <span>(infty )</span>-categories. There are three kinds of functors between duoidal <span>(infty )</span>-categories, which are called bilax, double lax, and double oplax monoidal functors. We make three formulations of <span>(infty )</span>-categories of duoidal <span>(infty )</span>-categories according to which functors we take. Furthermore, corresponding to the three kinds of functors, we define bimonoids, double monoids, and double comonoids in duoidal <span>(infty )</span>-categories.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"125 - 162"},"PeriodicalIF":0.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455434","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}
{"title":"On the homotopy type of partial quotients of certain moment-angle complexes","authors":"Xin Fu","doi":"10.1007/s40062-025-00363-y","DOIUrl":"10.1007/s40062-025-00363-y","url":null,"abstract":"<div><p>We consider moment-angle complexes associated with skeleta of simplices and determine the homotopy type of their quotient spaces under the diagonal circle action.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"105 - 123"},"PeriodicalIF":0.7,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455435","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}
{"title":"Classification of homogeneous functors in manifold calculus","authors":"Paul Arnaud Songhafouo Tsopméné, Donald Stanley","doi":"10.1007/s40062-025-00362-z","DOIUrl":"10.1007/s40062-025-00362-z","url":null,"abstract":"<div><p>For any object <i>A</i> in a simplicial model category <span>(mathcal {M})</span>, we construct a topological space <span>(hat{A})</span> which classifies homogeneous functors whose value on <i>k</i> open balls is equivalent to <i>A</i>. This extends a classification result of Weiss for homogeneous functors into topological spaces.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"63 - 103"},"PeriodicalIF":0.7,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455540","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}
{"title":"Dg Loday–Pirashvili modules over Lie algebras","authors":"Zhuo Chen, Yu Qiao, Maosong Xiang, Tao Zhang","doi":"10.1007/s40062-024-00361-6","DOIUrl":"10.1007/s40062-024-00361-6","url":null,"abstract":"<div><p>A Loday–Pirashvili module over a Lie algebra <span>(mathfrak {g})</span> is a Lie algebra object <span>(bigl (Gxrightarrow {X} mathfrak {g}bigr ))</span> in the category of linear maps, or equivalently, a <span>(mathfrak {g})</span>-module <i>G</i> which admits a <span>(mathfrak {g})</span>-equivariant linear map <span>(X:Grightarrow mathfrak {g})</span>. We study dg Loday–Pirashvili modules over Lie algebras, which is a generalization of Loday–Pirashvili modules in a natural way, and establish several equivalent characterizations of dg Loday–Pirashvili modules. To provide a concise characterization, a dg Loday–Pirashvili module is a non-negative and bounded dg <span>(mathfrak {g})</span>-module <i>V</i> paired with a weak morphism of dg <span>(mathfrak {g})</span>-modules <span>(alpha :Vrightsquigarrow mathfrak {g})</span>. Such a dg Loday–Pirashvili module resolves an arbitrarily specified classical Loday–Pirashvili module in the sense that it exists and is unique (up to homotopy). Dg Loday–Pirashvili modules can also be characterized through dg derivations. This perspective allows the calculation of the corresponding twisted Atiyah classes. By leveraging the Kapranov functor on the dg derivation arising from a dg Loday–Pirashvili module <span>((V,alpha ))</span>, a <span>(hbox {Leibniz}_infty [1])</span> algebra structure can be derived on <span>(wedge ^bullet mathfrak {g}^vee otimes V[1])</span>. The binary bracket of this structure corresponds to the twisted Atiyah cocycle. To exemplify these intricate algebraic structures through specific cases, we utilize this machinery to a particular type of dg Loday–Pirashvili modules stemming from Lie algebra pairs.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"23 - 61"},"PeriodicalIF":0.7,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455566","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}
{"title":"Sequential n-connectedness and infinite deformations of n-loops","authors":"Jeremy Brazas","doi":"10.1007/s40062-024-00360-7","DOIUrl":"10.1007/s40062-024-00360-7","url":null,"abstract":"<div><p>A space <i>X</i> is “sequentially <i>n</i>-connected” at <span>(xin X)</span> if for every <span>(0leqslant kleqslant n)</span> and sequence of <i>k</i>-loops <span>(f_1,f_2,f_3,ldots :S^krightarrow X)</span> that converges toward the point <i>x</i>, the maps <span>(f_m)</span> contract by a sequence of null-homotopies that converge toward <i>x</i>. Unlike standard local contractibility conditions, the sequential <i>n</i>-connectedness property is closed under forming infinite products and infinite shrinking wedges. We use this property, in conjunction with the Whitney Covering Lemma, to construct homotopies that simultaneously perform infinite deformations of <i>n</i>-loops and, ultimately, allow us to continuously deform arbitrary <i>n</i>-loops into maps with simpler forms. As a direct application, we extend the computation of the <i>n</i>-th homotopy group of a shrinking wedge of certain <span>((n-1))</span>-connected spaces due to K. Eda and K. Kawamura.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 1","pages":"1 - 22"},"PeriodicalIF":0.7,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455463","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}
{"title":"The Hurewicz model structure on simplicial R-modules","authors":"Arnaud Ngopnang Ngompé","doi":"10.1007/s40062-024-00359-0","DOIUrl":"10.1007/s40062-024-00359-0","url":null,"abstract":"<div><p>By a theorem of Christensen and Hovey, the category of non-negatively graded chain complexes has a model structure, called the h-model structure or Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. The Dold–Kan correspondence induces a model structure on the category of simplicial modules. In this paper, we give a description of the two model categories and some of their properties, notably the fact that both are monoidal.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"701 - 723"},"PeriodicalIF":0.7,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587755","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}
{"title":"The (mathbb {Z}/2) Fadell–Husseini index of the complex Grassmann manifolds (G_{n}(mathbb {C}^{2n}))","authors":"Arijit Nath, Avijit Nath","doi":"10.1007/s40062-024-00357-2","DOIUrl":"10.1007/s40062-024-00357-2","url":null,"abstract":"<div><p>In this paper, we study the <span>(mathbb {Z}/2)</span> action on complex Grassmann manifolds <span>(G_{n}(mathbb {C}^{2n}))</span> given by taking orthogonal complement. We completely compute the associated <span>(mathbb {Z}/2)</span> Fadell–Husseini index. Our study is parallel to the study of the index of real Grassmann manifolds <span>(G_n(mathbb {R}^{2n}))</span> by Baralić et al. [Forum Math., <b>30</b> (2018), pp. 1539–1572].</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"679 - 700"},"PeriodicalIF":0.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587754","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}
{"title":"Flat comodules and contramodules as directed colimits, and cotorsion periodicity","authors":"Leonid Positselski","doi":"10.1007/s40062-024-00358-1","DOIUrl":"10.1007/s40062-024-00358-1","url":null,"abstract":"<div><p>This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring <span>({mathcal {C}})</span> over a noncommutative ring <i>A</i>, we show that all <i>A</i>-flat <span>({mathcal {C}})</span>-comodules are <span>(aleph _1)</span>-directed colimits of <i>A</i>-countably presentable <i>A</i>-flat <span>({mathcal {C}})</span>-comodules. In the context of a complete, separated topological ring <span>({mathfrak {R}})</span> with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat <span>({mathfrak {R}})</span>-contramodules are <span>(aleph _1)</span>-directed colimits of countably presentable flat <span>({mathfrak {R}})</span>-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of <i>A</i>-flat <span>({mathcal {C}})</span>-comodules and flat <span>({mathfrak {R}})</span>-contramodules as <span>(aleph _1)</span>-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion <span>({mathfrak {R}})</span>-contramodules, all the contramodules of cocycles are cotorsion.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"635 - 678"},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00358-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587907","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}
{"title":"Lie 2-groups from loop group extensions","authors":"Matthias Ludewig, Konrad Waldorf","doi":"10.1007/s40062-024-00355-4","DOIUrl":"10.1007/s40062-024-00355-4","url":null,"abstract":"<div><p>We give a very simple construction of the string 2-group as a strict Fréchet Lie 2-group. The corresponding crossed module is defined using the conjugation action of the loop group on its central extension, which drastically simplifies several constructions previously given in the literature. More generally, we construct strict 2-group extensions for a Lie group from a central extension of its based loop group, under the assumption that this central extension is disjoint commutative. We show in particular that this condition is automatic in the case that the Lie group is semisimple and simply connected.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 4","pages":"597 - 633"},"PeriodicalIF":0.7,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00355-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587758","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}
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