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On Singer’s conjecture for the fourth algebraic transfer in certain generic degrees 关于辛格对某些通用度数中第四代数转移的猜想
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-07-13 DOI: 10.1007/s40062-024-00351-8
Ɖặng Võ Phúc
{"title":"On Singer’s conjecture for the fourth algebraic transfer in certain generic degrees","authors":"Ɖặng Võ Phúc","doi":"10.1007/s40062-024-00351-8","DOIUrl":"10.1007/s40062-024-00351-8","url":null,"abstract":"<div><p>Let <i>A</i> be the Steenrod algebra over the finite field <span>(k:= {mathbb {F}}_2)</span> and <i>G</i>(<i>q</i>) be the general linear group of rank <i>q</i> over <i>k</i>. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, <span>(textrm{Ext}^{q, *}_A(k, k),)</span> for all homological degrees <span>(q geqslant 0.)</span> The Singer algebraic transfer of rank <i>q</i>,  formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of <i>G</i>(<i>q</i>) to <span>(textrm{Ext}^{q, *}_A(k, k).)</span> Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all <span>(qgeqslant 4.)</span> This paper establishes Singer’s conjecture for rank four in the generic degrees <span>(n = 2^{s+t+1} +2^{s+1} - 3)</span> whenever <span>(tne 3)</span> and <span>(sgeqslant 1,)</span> and <span>(n = 2^{s+t} + 2^{s} - 2)</span> whenever <span>(tne 2,, 3,, 4)</span> and <span>(sgeqslant 1.)</span> In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"431 - 473"},"PeriodicalIF":0.7,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141613151","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
Enriched Koszul duality 丰富的科斯祖尔对偶性
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-07-03 DOI: 10.1007/s40062-024-00349-2
Björn Eurenius
{"title":"Enriched Koszul duality","authors":"Björn Eurenius","doi":"10.1007/s40062-024-00349-2","DOIUrl":"10.1007/s40062-024-00349-2","url":null,"abstract":"<div><p>We show that the category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras carry model structures compatible with their closed non-unital monoidal and closed non-unital module category structures respectively. Furthermore, we show that the Quillen equivalence between these two categories extends to a non-unital module category Quillen equivalence, i.e. providing an enriched form of Koszul duality.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"397 - 429"},"PeriodicalIF":0.7,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00349-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508677","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 mod 2 cohomology algebra of oriented Grassmannians 论面向格拉斯曼的模 2 同调代数
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-06-28 DOI: 10.1007/s40062-024-00350-9
Milica Jovanović, Branislav I. Prvulović
{"title":"On the mod 2 cohomology algebra of oriented Grassmannians","authors":"Milica Jovanović,&nbsp;Branislav I. Prvulović","doi":"10.1007/s40062-024-00350-9","DOIUrl":"10.1007/s40062-024-00350-9","url":null,"abstract":"<div><p>For <span>(nin {2^t-3,2^t-2,2^t-1})</span> <span>((tge 3))</span> we study the cohomology algebra <span>(H^*(widetilde{G}_{n,3};{mathbb {Z}}_2))</span> of the Grassmann manifold <span>(widetilde{G}_{n,3})</span> of oriented 3-dimensional subspaces of <span>({mathbb {R}}^n.)</span> A complete description of <span>(H^*(widetilde{G}_{n,3};{mathbb {Z}}_2))</span> is given in the cases <span>(n=2^t-3)</span> and <span>(n=2^t-2,)</span> while in the case <span>(n=2^t-1)</span> we obtain a description complete up to a coefficient from <span>({mathbb {Z}}_2.)</span></p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"379 - 396"},"PeriodicalIF":0.7,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529705","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
Two-sided cartesian fibrations of synthetic ((infty ,1))-categories 合成$$(infty ,1)$$-类的双侧笛卡尔纤度
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-06-21 DOI: 10.1007/s40062-024-00348-3
Jonathan Weinberger
{"title":"Two-sided cartesian fibrations of synthetic ((infty ,1))-categories","authors":"Jonathan Weinberger","doi":"10.1007/s40062-024-00348-3","DOIUrl":"10.1007/s40062-024-00348-3","url":null,"abstract":"<div><p>Within the framework of Riehl–Shulman’s synthetic <span>((infty ,1))</span>-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition à la Chevalley, Gray, Street, and Riehl–Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss <i>discrete</i> two-sided cartesian fibrations in this setting, corresponding to <span>((infty ,1))</span>-distributors. The systematics of our definitions and results closely follows Riehl–Verity’s <span>(infty )</span>-cosmos theory, but formulated internally to Riehl–Shulman’s simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic <span>((infty ,1))</span>-categories correspond to internal <span>((infty ,1))</span>-categories implemented as Rezk objects in an arbitrary given <span>((infty ,1))</span>-topos.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 3","pages":"297 - 378"},"PeriodicalIF":0.7,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00348-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508678","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
Multisimplicial chains and configuration spaces 多简链和构型空间
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-05-09 DOI: 10.1007/s40062-024-00344-7
Anibal M. Medina-Mardones, Andrea Pizzi, Paolo Salvatore
{"title":"Multisimplicial chains and configuration spaces","authors":"Anibal M. Medina-Mardones,&nbsp;Andrea Pizzi,&nbsp;Paolo Salvatore","doi":"10.1007/s40062-024-00344-7","DOIUrl":"10.1007/s40062-024-00344-7","url":null,"abstract":"<div><p>We define an <span>(E_infty )</span>-coalgebra structure on the chains of multisimplicial sets. Our primary focus is on the surjection chain complexes of McClure-Smith, for which we construct a zig-zag of complexity preserving quasi-isomorphisms of <span>(E_infty )</span>-coalgebras relating them to both the singular chains on configuration spaces and the Barratt–Eccles chain complexes.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"275 - 296"},"PeriodicalIF":0.7,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-024-00344-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930170","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
Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2 透镜空间上极性莫尔斯-波特叶形的差分群的同调类型，2
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-04-18 DOI: 10.1007/s40062-024-00346-5
Sergiy Maksymenko
{"title":"Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 2","authors":"Sergiy Maksymenko","doi":"10.1007/s40062-024-00346-5","DOIUrl":"10.1007/s40062-024-00346-5","url":null,"abstract":"<div><p>Let <span>({mathcal {F}})</span> be a Morse–Bott foliation on the solid torus <span>(T=S^1times D^2)</span> into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of <i>T</i> by some diffeomorphism between their boundaries, one gets a lens space <span>(L_{p,q})</span> with a Morse–Bott foliation <span>({mathcal {F}}_{p,q})</span> obtained from <span>({mathcal {F}})</span> on each copy of <i>T</i> and thus consisting of two singular circles and parallel 2-tori. In the previous paper Khokliuk and Maksymenko (J Homotopy Relat Struct 18:313–356. https://doi.org/10.1007/s40062-023-00328-z, 2024) there were computed weak homotopy types of the groups <span>({mathcal {D}}^{lp}({mathcal {F}}_{p,q}))</span> of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group <span>({mathcal {D}}^{fol}_{+}({mathcal {F}}_{p,q}))</span> of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"239 - 273"},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140629859","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
Diffeological principal bundles and principal infinity bundles 差分主束和主无穷束
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-04-15 DOI: 10.1007/s40062-024-00347-4
Emilio Minichiello
{"title":"Diffeological principal bundles and principal infinity bundles","authors":"Emilio Minichiello","doi":"10.1007/s40062-024-00347-4","DOIUrl":"10.1007/s40062-024-00347-4","url":null,"abstract":"<div><p>In this paper, we study diffeological spaces as certain kinds of discrete simplicial presheaves on the site of cartesian spaces with the coverage of good open covers. The Čech model structure on simplicial presheaves provides us with a notion of <span>(infty )</span>-stack cohomology of a diffeological space with values in a diffeological abelian group <i>A</i>. We compare <span>(infty )</span>-stack cohomology of diffeological spaces with two existing notions of Čech cohomology for diffeological spaces in the literature Krepski et al. (Sheaves, principal bundles, and Čech cohomology for diffeological spaces. (2021). arxiv:2111 01032 [math.DG]), Iglesias-Zemmour (Čech-de-Rham Bicomplex in Diffeology (2020). http://math.huji.ac.il/piz/documents/CDRBCID.pdf). Finally, we prove that for a diffeological group <i>G</i>, that the nerve of the category of diffeological principal <i>G</i>-bundles is weak homotopy equivalent to the nerve of the category of <i>G</i>-principal <span>(infty )</span>-bundles on <i>X</i>, bridging the bundle theory of diffeology and higher topos theory.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"181 - 237"},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560997","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 Matsumoto type theorem for (GL_n) over rings of non-commutative Laurent polynomials 非交换月桂多项式环上 $$GL_n$$ 的松本类型定理
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-04-06 DOI: 10.1007/s40062-024-00345-6
Ryusuke Sugawara
{"title":"A Matsumoto type theorem for (GL_n) over rings of non-commutative Laurent polynomials","authors":"Ryusuke Sugawara","doi":"10.1007/s40062-024-00345-6","DOIUrl":"10.1007/s40062-024-00345-6","url":null,"abstract":"<div><p>We give a Matsumoto-type presentation of <span>(K_2)</span>-groups over rings of non-commutative Laurent polynomials, which is a non-commutative version of M. Tomie’s result for loop groups. Our main idea is induced by U. Rehmann’s approach in the case of division rings.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"151 - 180"},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561000","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 reasonable notion of dimension for singular intersection homology 奇异交点同调的合理维度概念
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-04-04 DOI: 10.1007/s40062-024-00343-8
David Chataur, Martintxo Saralegi-Aranguren, Daniel Tanré
{"title":"A reasonable notion of dimension for singular intersection homology","authors":"David Chataur,&nbsp;Martintxo Saralegi-Aranguren,&nbsp;Daniel Tanré","doi":"10.1007/s40062-024-00343-8","DOIUrl":"10.1007/s40062-024-00343-8","url":null,"abstract":"<div><p>M. Goresky and R. MacPherson intersection homology is also defined from the singular chain complex of a filtered space by H. King, with a key formula to make selections among singular simplexes. This formula needs a notion of dimension for subspaces <i>S</i> of an Euclidean simplex, which is usually taken as the smallest dimension of the skeleta containing <i>S</i>. Later, P. Gajer employed another dimension based on the dimension of polyhedra containing <i>S</i>. This last one allows traces of pullbacks of singular strata in the interior of the domain of a singular simplex. In this work, we prove that the two corresponding intersection homologies are isomorphic for Siebenmann’s CS sets. In terms of King’s paper, this means that polyhedral dimension is a “reasonable” dimension. The proof uses a Mayer-Vietoris argument which needs an adapted subdivision. With the polyhedral dimension, that is a subtle issue. General position arguments are not sufficient and we introduce strong general position. With it, a stability is added to the generic character and we can do an inductive cutting of each singular simplex. This decomposition is realised with pseudo-barycentric subdivisions where the new vertices are not barycentres but close points of them.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"121 - 150"},"PeriodicalIF":0.7,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560999","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
Adams operations on the twisted K-theory of compact Lie groups 紧凑李群扭曲 K 理论上的亚当斯运算
IF 0.7 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2024-03-18 DOI: 10.1007/s40062-024-00342-9
Chi-Kwong Fok
{"title":"Adams operations on the twisted K-theory of compact Lie groups","authors":"Chi-Kwong Fok","doi":"10.1007/s40062-024-00342-9","DOIUrl":"10.1007/s40062-024-00342-9","url":null,"abstract":"<div><p>In this paper, extending the results in Fok (Proc Am Math Soc 145:2799–2813, 2017), we compute Adams operations on the twisted <i>K</i>-theory of connected, simply-connected and simple compact Lie groups <i>G</i>, in both equivariant and nonequivariant settings.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"19 2","pages":"99 - 120"},"PeriodicalIF":0.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149962","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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