{"title":"Realization of saturated transfer systems on cyclic groups of order (p^nq^m) by linear isometries (N_infty )-operads","authors":"Julie Bannwart","doi":"10.1007/s40062-025-00377-6","DOIUrl":"10.1007/s40062-025-00377-6","url":null,"abstract":"<div><p>We prove a specific case of Rubin’s saturation conjecture about the realization of <i>G</i>-transfer systems, for <i>G</i> a finite cyclic group, by linear isometries <span>(N_infty )</span>-operads, namely the case of cyclic groups of order <span>(p^nq^m)</span> for <i>p</i>, <i>q</i> distinct primes and <span>(n,min mathbb {N})</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 3","pages":"455 - 475"},"PeriodicalIF":0.5,"publicationDate":"2025-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00377-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810903","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":"Explicit sharbly cycles at the virtual cohomological dimension for (textrm{SL}_n(mathbb {Z}))","authors":"Avner Ash, Paul E. Gunnells, Mark McConnell","doi":"10.1007/s40062-025-00374-9","DOIUrl":"10.1007/s40062-025-00374-9","url":null,"abstract":"<div><p>Denote the virtual cohomological dimension of <span>(textrm{SL}_n(mathbb {Z}))</span> by <span>(t=n(n-1)/2)</span>. Let <i>St</i> denote the Steinberg module of <span>(textrm{SL}_n(mathbb {Q}))</span> tensored with <span>(mathbb {Q})</span>. Let <span>(Sh_bullet rightarrow St)</span> denote the sharbly resolution of the Steinberg module. By Borel–Serre duality, the one-dimensional <span>(mathbb {Q})</span>-vector space <span>(H^0(textrm{SL}_n(mathbb {Z}), mathbb {Q}))</span> is isomorphic to <span>(H_t(textrm{SL}_n(mathbb {Z}),St))</span>. We find an explicit generator of <span>(H_t(textrm{SL}_n(mathbb {Z}),St))</span> in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of <span>(textrm{SL}_n(mathbb {Z}))</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 3","pages":"391 - 416"},"PeriodicalIF":0.5,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810812","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":"Higher (equivariant) topological complexity of Milnor manifolds","authors":"Navnath Daundkar, Bittu Singh","doi":"10.1007/s40062-025-00376-7","DOIUrl":"10.1007/s40062-025-00376-7","url":null,"abstract":"<div><p>J. Milnor introduced a specific class of codimension-1 submanifolds in the product of projective spaces, known as Milnor manifolds. This paper establishes precise bounds on the higher topological complexity of these manifolds and provides exact values for this invariant for numerous Milnor manifolds. Furthermore, we improve the upper bounds on the higher equivariant topological complexity. As an application, we obtain sharper bounds on the higher equivariant topological complexity of Milnor manifolds with free <span>(mathbb {Z}_2)</span> and <span>(S^1)</span>-actions.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 3","pages":"437 - 453"},"PeriodicalIF":0.5,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810771","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":"Bigraded Poincaré polynomials and the equivariant cohomology of Rep((C_2))-complexes","authors":"Eric Hogle","doi":"10.1007/s40062-025-00375-8","DOIUrl":"10.1007/s40062-025-00375-8","url":null,"abstract":"<div><p>We are interested in computing the Bredon cohomology with coefficients in the constant Mackey functor <span>(underline{{mathbb {F}}_2})</span> for equivariant <span>(text {Rep}(C_2))</span> spaces, in particular for Grassmannian manifolds of the form <span>(operatorname {Gr}_k(V))</span> where <i>V</i> is some real representation of <span>(C_2.)</span> It is possible to create multiple distinct <span>(text {Rep}(C_2))</span> constructions of (and hence multiple filtration spectral sequences for) a given Grassmannian. For sufficiently small examples one may exhaustively compute all possible outcomes of each spectral sequence and determine if there exists a unique common answer. However, the complexity of such a computation quickly balloons in time and memory requirements. We introduce a statistic on <span>(mathbb {M}_2)</span>-modules valued in the polynomial ring <span>(mathbb Z[x,y])</span> which makes cohomology computation of Rep<span>((C_2))</span>-complexes more tractable, and we present some new results for Grassmannians.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 3","pages":"417 - 435"},"PeriodicalIF":0.5,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810700","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":"Cosimplicial structure on pointed multiplicative operads","authors":"V. Jacky III Batkam Mbatchou, Calvin Tcheka","doi":"10.1007/s40062-025-00372-x","DOIUrl":"10.1007/s40062-025-00372-x","url":null,"abstract":"<div><p>Motivated by the work of Gerstenhaber-Voronov and that of Malvenuto-Reuternauer, we define on pointed multiplicative operads in the category of vector spaces over an arbitrary ground field <span>(mathbb {K})</span>, a cosimplicial vector space structure. This permits us to construct on such operads some algebraic structures such as the homotopy G-algebra and the bicomplex algebra structures. Moreover we illustrate our constructions through some examples and explain or extend some well-known results.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 3","pages":"365 - 385"},"PeriodicalIF":0.5,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810826","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":"Concerning monoid structures on naive homotopy classes of endomorphisms of punctured affine space","authors":"Thomas Brazelton, William Hornslien","doi":"10.1007/s40062-025-00373-w","DOIUrl":"10.1007/s40062-025-00373-w","url":null,"abstract":"<div><p>Cazanave proved that the set of naive <span>(mathbb {A}^1)</span>-homotopy classes of endomorphisms of the projective line admits a monoid structure whose group completion is genuine <span>(mathbb {A}^1)</span>-homotopy classes of endomorphisms of the projective line. In this very short note we show that, over a field which is not quadratically closed, such a statement is never true for punctured affine space <span>(mathbb {A}^nhspace{-0.1em}smallsetminus {0})</span> for <span>(nge 2)</span>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 3","pages":"387 - 390"},"PeriodicalIF":0.5,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810919","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 real cycle class map for singular varieties","authors":"Fangzhou Jin, Heng Xie","doi":"10.1007/s40062-025-00369-6","DOIUrl":"10.1007/s40062-025-00369-6","url":null,"abstract":"<div><p>We investigate the real cycle class map for singular varieties. We introduce an analog of Borel–Moore homology for algebraic varieties over the real numbers, which is defined via the hypercohomology of the Gersten–Witt complex associated with schemes possessing a dualizing complex. We show that the hypercohomology of this complex is isomorphic to the classical Borel–Moore homology for quasi-projective varieties over the real numbers.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 2","pages":"293 - 321"},"PeriodicalIF":0.7,"publicationDate":"2025-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00369-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143925691","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":"Localisations and completions of nilpotent G-spaces","authors":"Andrew Ronan","doi":"10.1007/s40062-025-00371-y","DOIUrl":"10.1007/s40062-025-00371-y","url":null,"abstract":"<div><p>We develop the theory of nilpotent <i>G</i>-spaces and their localisations, for <i>G</i> a compact Lie group, via reduction to the non-equivariant case using Bousfield localisation. One point of interest in the equivariant setting is that we can choose to localise or complete at different sets of primes at different fixed point spaces—and the theory works out just as well provided that you invert more primes at <span>(K le G)</span> than at <span>(H le G)</span>, whenever <i>K</i> is subconjugate to <i>H</i> in <i>G</i>. We also develop the theory in an unbased context, allowing us to extend the theory to <i>G</i>-spaces which are not <i>G</i>-connected.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 2","pages":"331 - 364"},"PeriodicalIF":0.7,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00371-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143925550","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":"A conjecture on the composition of localizations on a stratified tensor triangulated category","authors":"Nicola Bellumat","doi":"10.1007/s40062-025-00367-8","DOIUrl":"10.1007/s40062-025-00367-8","url":null,"abstract":"<div><p>We study the composition of Bousfield localizations on a tensor triangulated category stratified via the Balmer-Favi support and with noetherian Balmer spectrum. Our aim is to provide reductions via purely axiomatic arguments, allowing us general applications to concrete categories examined in mathematical practice. We propose a conjecture which states that the behaviour of the composition of the localizations depends on the chains of inclusions of the Balmer primes indexing said localizations. We prove this conjecture in the case of finite or low dimensional Balmer spectra.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 2","pages":"251 - 285"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00367-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143925518","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":"On split steinberg modules and steinberg modules","authors":"Daniel Armeanu, Jeremy Miller","doi":"10.1007/s40062-025-00370-z","DOIUrl":"10.1007/s40062-025-00370-z","url":null,"abstract":"<div><p>Answering a question of Randal-Williams, we show the natural maps from split Steinberg modules of a Dedekind domain to the associated Steinberg modules are surjective.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"20 2","pages":"323 - 329"},"PeriodicalIF":0.7,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-025-00370-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143925519","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}