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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
Hodge decompositions and differential Poincaré duality models Hodge分解与微分poincarcarcarr对偶模型
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-12-22 DOI: 10.1007/s40062-025-00389-2
Pavel Hájek
{"title":"Hodge decompositions and differential Poincaré duality models","authors":"Pavel Hájek","doi":"10.1007/s40062-025-00389-2","DOIUrl":"10.1007/s40062-025-00389-2","url":null,"abstract":"<div><p>We extend a CDGA <i>V</i> with a perfect pairing of degree <i>n</i> on cohomology to a CDGA <span>(hat{V})</span> with a pairing of degree <i>n</i> on chain level such that <span>(hat{V})</span> admits a Hodge decomposition and retracts onto <i>V</i> preserving the pairing on cohomology; here we suppose that <i>V</i> is either 1-connected, or that <i>V</i> is connected, of finite type, and <i>n</i> is odd. We show that a Hodge decomposition of <span>(hat{V})</span> induces a differential Poincaré duality model of <i>V</i> in a natural way. Assuming that <span>(textrm{H}(V))</span> is 1-connected, we apply our extension to a Sullivan model of <i>V</i> in the proof of the existence and “uniqueness” of a 1-connected differential Poincaré duality model of <i>V</i> by Lambrechts &amp; Stanley; we eliminate their extra assumptions in the uniqueness statement, including <span>(textrm{H}^2(V)=0)</span> if <i>n</i> is odd.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 2","pages":"175 - 210"},"PeriodicalIF":0.5,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00389-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147727451","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 analytic exponential functors on free groups 关于自由群上的解析指数函子
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-12-03 DOI: 10.1007/s40062-025-00390-9
Minkyu Kim, Christine Vespa
{"title":"On analytic exponential functors on free groups","authors":"Minkyu Kim,&nbsp;Christine Vespa","doi":"10.1007/s40062-025-00390-9","DOIUrl":"10.1007/s40062-025-00390-9","url":null,"abstract":"<div><p>This paper concerns exponential contravariant functors on free groups. We obtain an equivalence of categories between analytic, exponential contravariant functors on free groups and conilpotent cocommutative Hopf algebras. This result explains how equivalences of categories obtained previously by Pirashvili and by Powell interact. Moreover, we obtain an equivalence between the categories of outer, exponential contravariant functors on free groups and bicommutative Hopf algebras. We also go further by introducing a subclass of analytic, contravariant functors on free groups, called <i>primitive</i> functors; and prove an equivalence between primitive, exponential contravariant functors and primitive cocommutative Hopf algebras.\u0000</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 2","pages":"131 - 174"},"PeriodicalIF":0.5,"publicationDate":"2025-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147727340","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 equivariant covering homotopy property 等变覆盖同伦性质
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-11-26 DOI: 10.1007/s40062-025-00392-7
Andrew Ronan
{"title":"The equivariant covering homotopy property","authors":"Andrew Ronan","doi":"10.1007/s40062-025-00392-7","DOIUrl":"10.1007/s40062-025-00392-7","url":null,"abstract":"<div><p>In this paper, we explain how the more general context of generalised equivariant bundles allows for a simple proof of the ECHP, which makes use of induction on the dimension/no. of connected components of compact Lie groups. We also make clear the link between the ECHP and the theory of Hurewicz fibrations.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 1","pages":"117 - 129"},"PeriodicalIF":0.5,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147341915","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 conceptual derivation of the dual Steenrod algebra 对偶Steenrod代数的概念推导
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-11-26 DOI: 10.1007/s40062-025-00391-8
Kiran Luecke
{"title":"A conceptual derivation of the dual Steenrod algebra","authors":"Kiran Luecke","doi":"10.1007/s40062-025-00391-8","DOIUrl":"10.1007/s40062-025-00391-8","url":null,"abstract":"<div><p>In this note I give a conceptual proof of the fact that the mod 2 dual Steenrod algebra corepresents the group scheme of strict automorphisms of the formal additive group over <span>({mathbb {F}}_2)</span>. Contrary to existing proofs, it does not use the <span>(E_infty )</span>-structure of <span>(H{mathbb {F}}_2)</span> (Steenrod operations), nor does it proceed by producing a generators-and-relations presentation by some explicit calculation. Instead it relies on universal properties of bordism spectra, thus giving a stronger conceptual foundation for what is arguably the first instance of the well-studied deep connection between the algebraic geometry of formal groups and the stable homotopy category.\u0000</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 1","pages":"107 - 116"},"PeriodicalIF":0.5,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00391-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147341914","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 the topology of (mathcal {M}_{0,n+1}/Sigma _n) 的拓扑结构 (mathcal {M}_{0,n+1}/Sigma _n)
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-11-02 DOI: 10.1007/s40062-025-00388-3
Tommaso Rossi
{"title":"On the topology of (mathcal {M}_{0,n+1}/Sigma _n)","authors":"Tommaso Rossi","doi":"10.1007/s40062-025-00388-3","DOIUrl":"10.1007/s40062-025-00388-3","url":null,"abstract":"<div><p>This paper contains some results about the topology of <span>(mathcal {M}_{0,n+1}/Sigma _n)</span>, where <span>(mathcal {M}_{0,n+1})</span> is the moduli space of genus zero Riemann surfaces with marked points. We show that <span>(mathcal {M}_{0,n+1}/Sigma _n)</span> is not a topological manifold for <span>(nge 4)</span>, and it is simply connected for any <span>(nin mathbb {N})</span>. We also present some homology computations: for example we show that <span>(mathcal {M}_{0,p+1}/Sigma _p)</span> has no <i>p</i> torsion, where <i>p</i> is a prime. Lastly we compute <span>(H_*(mathcal {M}_{0,n+1}/Sigma _n;mathbb {Z}))</span> for small values of <i>n</i>, proving that <span>(mathcal {M}_{0,n+1}/Sigma _n)</span> is contractible for <span>(nle 5)</span> while <span>(mathcal {M}_{0,7}/Sigma _6)</span> is not.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 1","pages":"81 - 106"},"PeriodicalIF":0.5,"publicationDate":"2025-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00388-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147335789","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
Constructing Reedy fibrant replacements of projective fibrant simplicial presheaves 构造芦苇纤维替换投影纤维简单预轴
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-10-25 DOI: 10.1007/s40062-025-00386-5
Jack Romö
{"title":"Constructing Reedy fibrant replacements of projective fibrant simplicial presheaves","authors":"Jack Romö","doi":"10.1007/s40062-025-00386-5","DOIUrl":"10.1007/s40062-025-00386-5","url":null,"abstract":"<div><p>In this paper, we construct an explicit Reedy fibrant replacement functor for projective fibrant simplicial presheaves <span>(X : mathscr {C} rightarrow {textbf {sSet}})</span>, where <span>(mathscr {C})</span> is a Reedy category. Our approach describes, by hand, all latching maps for the Reedy fibrant replacement by an inductive series of higher homotopies. The concrete nature of the construction means it has application in extending other constructions on Reedy fibrant functors to more general projective fibrant functors, providing an explicit description of the extension. In particular, the author uses the Reedy fibrant replacement functor presented here in his thesis to construct the homotopy bicategory of a projective fibrant 2-fold Segal space by extending a similar construction in the Reedy fibrant case. We illustrate our functor’s behavior by using it to recover the homotopy category of a projective fibrant Segal space from the classical homotopy category construction for Reedy fibrant Segal spaces, which allows us to further recover a standard characterization of completeness for projective fibrant Segal spaces.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 1","pages":"45 - 80"},"PeriodicalIF":0.5,"publicationDate":"2025-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00386-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147341701","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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