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Higher (equivariant) topological complexity of Milnor manifolds 米尔诺流形的高（等变）拓扑复杂性
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-07-01 DOI: 10.1007/s40062-025-00376-7
Navnath Daundkar, Bittu Singh
{"title":"Higher (equivariant) topological complexity of Milnor manifolds","authors":"Navnath Daundkar,&nbsp;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}
引用次数: 0
Bigraded Poincaré polynomials and the equivariant cohomology of Rep((C_2))-complexes Rep ((C_2)) -配合物的重阶poincarcars多项式和等变上同调
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-06-27 DOI: 10.1007/s40062-025-00375-8
Eric Hogle
{"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}
引用次数: 0
Cosimplicial structure on pointed multiplicative operads 点乘法操作数上的共简结构
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-06-17 DOI: 10.1007/s40062-025-00372-x
V. Jacky III Batkam Mbatchou, Calvin Tcheka
{"title":"Cosimplicial structure on pointed multiplicative operads","authors":"V. Jacky III Batkam Mbatchou,&nbsp;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}
引用次数: 0
Concerning monoid structures on naive homotopy classes of endomorphisms of punctured affine space 刺穿仿射空间中自同态的朴素同伦类上的单群结构
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-06-16 DOI: 10.1007/s40062-025-00373-w
Thomas Brazelton, William Hornslien
{"title":"Concerning monoid structures on naive homotopy classes of endomorphisms of punctured affine space","authors":"Thomas Brazelton,&nbsp;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}
引用次数: 0
On the real cycle class map for singular varieties 关于奇异变量的实循环类映射
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-04-21 DOI: 10.1007/s40062-025-00369-6
Fangzhou Jin, Heng Xie
{"title":"On the real cycle class map for singular varieties","authors":"Fangzhou Jin,&nbsp;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}
引用次数: 0
Localisations and completions of nilpotent G-spaces 幂零g空间的局部化与补全
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-04-11 DOI: 10.1007/s40062-025-00371-y
Andrew Ronan
{"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}
引用次数: 0
A conjecture on the composition of localizations on a stratified tensor triangulated category 关于分层张量三角化范畴上局部化组成的猜想
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-04-09 DOI: 10.1007/s40062-025-00367-8
Nicola Bellumat
{"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}
引用次数: 0
On split steinberg modules and steinberg modules 关于拆分斯坦伯格模块和斯坦伯格模块
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-04-09 DOI: 10.1007/s40062-025-00370-z
Daniel Armeanu, Jeremy Miller
{"title":"On split steinberg modules and steinberg modules","authors":"Daniel Armeanu,&nbsp;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}
引用次数: 0
A localization theorem for cyclic equivariant K-theory 循环等变k理论的一个局部化定理
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-04-04 DOI: 10.1007/s40062-025-00368-7
Jack Carlisle
{"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}
引用次数: 0
Normed symmetric monoidal categories 赋范对称单一性范畴
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2025-02-17 DOI: 10.1007/s40062-025-00366-9
Jonathan Rubin
{"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}
引用次数: 0
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