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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
The stable Picard group of (mathcal {A}(n)) 稳定的皮卡德群 (mathcal {A}(n))
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-10-24 DOI: 10.1007/s40062-025-00387-4
JianZhong Pan, RuJia Yan
{"title":"The stable Picard group of (mathcal {A}(n))","authors":"JianZhong Pan,&nbsp;RuJia Yan","doi":"10.1007/s40062-025-00387-4","DOIUrl":"10.1007/s40062-025-00387-4","url":null,"abstract":"<div><p>In this paper, we show that, for <span>(nge 2)</span>, the stable Picard group of <span>(mathcal {A}(n))</span> is <span>(mathbb {Z}oplus mathbb {Z})</span>, where <span>(mathcal {A}(n))</span> is the usual finite sub Hopf algebra of the Steenrod algebra <span>(mathcal {A})</span> at the prime 2. The proof relies on reductions from a Hopf algebra to certain sub Hopf algebras.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 1","pages":"23 - 43"},"PeriodicalIF":0.5,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147341922","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
Cohomology at infinity and the well-tempered complex 无穷远上同调与均匀复形
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-10-18 DOI: 10.1007/s40062-025-00385-6
Dylan Galt, Mark McConnell
{"title":"Cohomology at infinity and the well-tempered complex","authors":"Dylan Galt,&nbsp;Mark McConnell","doi":"10.1007/s40062-025-00385-6","DOIUrl":"10.1007/s40062-025-00385-6","url":null,"abstract":"<div><p>We prove the existence of a sequence of commutative diagrams generalizing the results on cohomology at infinity described in Ash and McConnell (Duke Math J 90:549–576, 1997) to the context of the well-tempered complex introduced in McConnell and MacPherson (Computing Hecke operators for arithmetic subgroups of general linear groups, http://arxiv.org/abs/2010.06036, 2020). Our main theorem provides a method for computing in finite terms the action of Hecke operators on the cohomology of the Borel-Serre boundary for the <span>(text {SL} _n)</span> symmetric space.\u0000</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"21 1","pages":"1 - 21"},"PeriodicalIF":0.5,"publicationDate":"2025-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339954","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
Morava K-theory rings for finite groups 有限群的Morava k -理论环
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-10-16 DOI: 10.1007/s40062-025-00384-7
Malkhaz Bakuradze
{"title":"Morava K-theory rings for finite groups","authors":"Malkhaz Bakuradze","doi":"10.1007/s40062-025-00384-7","DOIUrl":"10.1007/s40062-025-00384-7","url":null,"abstract":"<div><p>This paper compiles and expands upon the author’s and his co-authors’ explicit calculations of the mod <i>p</i> Morava K-theory for various finite <i>p</i>-groups, a body of work currently scattered across different publications. The primary focus is on the author’s observations regarding the properties of formal group laws and the transfer in Morava K-theory. Using specific examples, this work aims to clarify the complex issues surrounding the multiplicative structure and the representation of Gröbner bases in terms of Chern classes and their transfers. A key computational question remains: is the mod 2 Morava K-theory of any finite 2-group completely generated by Chern classes and their transfers? While this conjecture by Hopkins et al. (J Am Math Soc 13:553–594, 2000), inspired by their generalized character theory of finite <i>p</i>-groups, was disproven for the mod <span>(p&gt;2)</span> case by a counterexample in Kriz (Topology 36:1247–1273, 1997), the mod 2 case remains an open problem.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 4","pages":"567 - 630"},"PeriodicalIF":0.5,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145429131","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
Revisiting the Nandakumar–Ramana Rao conjecture 重新审视Nandakumar-Ramana Rao猜想
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-09-09 DOI: 10.1007/s40062-025-00383-8
Surojit Ghosh, Ankit Kumar
{"title":"Revisiting the Nandakumar–Ramana Rao conjecture","authors":"Surojit Ghosh,&nbsp;Ankit Kumar","doi":"10.1007/s40062-025-00383-8","DOIUrl":"10.1007/s40062-025-00383-8","url":null,"abstract":"<div><p>We reprove the generalized Nandakumar–Ramana Rao conjecture for the prime case using representation ring-graded Bredon cohomology. Our approach relies solely on the <span>(RO(C_p))</span>-graded cohomology of configuration spaces, viewed as a module over the <span>(RO(C_p))</span>-graded Bredon cohomology of a point.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 4","pages":"555 - 565"},"PeriodicalIF":0.5,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145429110","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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