{"title":"A localization theorem for cyclic equivariant K-theory","authors":"Jack Carlisle","doi":"10.1007/s40062-025-00368-7","DOIUrl":"10.1007/s40062-025-00368-7","url":null,"abstract":"<div><p>For a finite cyclic group <span>(C_n)</span>, we identify Greenlees’ equivariant connective K-theory <span>(kU_{C_n})</span> as an <span>(RO(C_n))</span>-graded localization of the actual connective cover of <span>(KU_{C_n})</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 2","pages":"287 - 292"},"PeriodicalIF":0.7,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00368-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143925541","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":"Normed symmetric monoidal categories","authors":"Jonathan Rubin","doi":"10.1007/s40062-025-00366-9","DOIUrl":"10.1007/s40062-025-00366-9","url":null,"abstract":"<div><p>We introduce categorical models of <span>(N_infty )</span> spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a particular class of examples, they reveal a connection between the equivariant symmetric monoidal categories of Guillou–May–Merling–Osorno and those of Hill–Hopkins. We also give an operadic interpretation of the Mac Lane coherence theorem and generalize it to include NSMCs. Among other things, this theorem ensures that the classifying space of an NSMC is an <span>(N_infty )</span> space. We conclude by extending our coherence theorem to include NSMCs with strict relations.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 2","pages":"195 - 250"},"PeriodicalIF":0.7,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143925610","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":"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}
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