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The completion theorem in twisted equivariant K-theory for proper actions 固有作用的扭曲等变k理论中的补全定理
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-01-31 DOI: 10.1007/s40062-021-00299-z
Noé Bárcenas, Mario Velásquez
{"title":"The completion theorem in twisted equivariant K-theory for proper actions","authors":"Noé Bárcenas,&nbsp;Mario Velásquez","doi":"10.1007/s40062-021-00299-z","DOIUrl":"10.1007/s40062-021-00299-z","url":null,"abstract":"<div><p>We compare different algebraic structures in twisted equivariant <i>K</i>-theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-theory, we prove a completion Theorem of Atiyah–Segal type for twisted equivariant K-theory. Using a universal coefficient theorem, we prove a cocompletion Theorem for twisted Borel K-homology for discrete groups.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 1","pages":"77 - 104"},"PeriodicalIF":0.5,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5172963","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
(C_2)-equivariant topological modular forms (C_2)-等变拓扑模形式
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-01-10 DOI: 10.1007/s40062-021-00297-1
Dexter Chua
{"title":"(C_2)-equivariant topological modular forms","authors":"Dexter Chua","doi":"10.1007/s40062-021-00297-1","DOIUrl":"10.1007/s40062-021-00297-1","url":null,"abstract":"<div><p>We compute the homotopy groups of the <span>(C_2)</span> fixed points of equivariant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a <span>({mathrm {TMF}})</span>-module, it is isomorphic to the tensor product of <span>({mathrm {TMF}})</span> with an explicit finite cell complex.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 1","pages":"23 - 75"},"PeriodicalIF":0.5,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4418988","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: 1
Marked colimits and higher cofinality 界限明显，共通性高
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-12-16 DOI: 10.1007/s40062-021-00296-2
Fernando Abellán García
{"title":"Marked colimits and higher cofinality","authors":"Fernando Abellán García","doi":"10.1007/s40062-021-00296-2","DOIUrl":"10.1007/s40062-021-00296-2","url":null,"abstract":"<div><p>Given a marked <span>(infty )</span>-category <span>(mathcal {D}^{dagger })</span> (i.e. an <span>(infty )</span>-category equipped with a specified collection of morphisms) and a functor <span>(F: mathcal {D}rightarrow {mathbb {B}})</span> with values in an <span>(infty )</span>-bicategory, we define <img>, the marked colimit of <i>F</i>. We provide a definition of weighted colimits in <span>(infty )</span>-bicategories when the indexing diagram is an <span>(infty )</span>-category and show that they can be computed in terms of marked colimits. In the maximally marked case <span>(mathcal {D}^{sharp })</span>, our construction retrieves the <span>(infty )</span>-categorical colimit of <i>F</i> in the underlying <span>(infty )</span>-category <span>(mathcal {B}subseteq {mathbb {B}})</span>. In the specific case when <img>, the <span>(infty )</span>-bicategory of <span>(infty )</span>-categories and <span>(mathcal {D}^{flat })</span> is minimally marked, we recover the definition of lax colimit of Gepner–Haugseng–Nikolaus. We show that a suitable <span>(infty )</span>-localization of the associated coCartesian fibration <span>({text {Un}}_{mathcal {D}}(F))</span> computes <img>. Our main theorem is a characterization of those functors of marked <span>(infty )</span>-categories <span>({f:mathcal {C}^{dagger } rightarrow mathcal {D}^{dagger }})</span> which are marked cofinal. More precisely, we provide sufficient and necessary criteria for the restriction of diagrams along <i>f</i> to preserve marked colimits</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 1","pages":"1 - 22"},"PeriodicalIF":0.5,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-021-00296-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4640649","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: 1
On the LS-category and topological complexity of projective product spaces 关于射影积空间的ls -范畴和拓扑复杂度
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-11-08 DOI: 10.1007/s40062-021-00295-3
Seher Fişekci, Lucile Vandembroucq
{"title":"On the LS-category and topological complexity of projective product spaces","authors":"Seher Fişekci,&nbsp;Lucile Vandembroucq","doi":"10.1007/s40062-021-00295-3","DOIUrl":"10.1007/s40062-021-00295-3","url":null,"abstract":"<div><p>We determine the Lusternik-Schnirelmann category of the projective product spaces introduced by D. Davis. We also obtain an upper bound for the topological complexity of these spaces, which improves the estimate given by J. González, M. Grant, E. Torres-Giese, and M. Xicoténcatl.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 4","pages":"769 - 780"},"PeriodicalIF":0.5,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-021-00295-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4353260","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: 4
Overcategories and undercategories of cofibrantly generated model categories 共同生成的模型类别的超类别和下类别
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-10-13 DOI: 10.1007/s40062-021-00294-4
Philip S. Hirschhorn
{"title":"Overcategories and undercategories of cofibrantly generated model categories","authors":"Philip S. Hirschhorn","doi":"10.1007/s40062-021-00294-4","DOIUrl":"10.1007/s40062-021-00294-4","url":null,"abstract":"<div><p>Let <span>(mathcal {M})</span> be a model category and let <i>Z</i> be an object of <span>(mathcal {M})</span>. We show that if <span>(mathcal {M})</span> is cofibrantly generated, cellular, left proper, or right proper, then both the model category <img> of objects of <span>(mathcal {M})</span> over <i>Z</i> and the model category <img> of objects of <span>(mathcal {M})</span> under <i>Z</i> are as well.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 4","pages":"753 - 768"},"PeriodicalIF":0.5,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4556503","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: 6
Rational model for the string coproduct of pure manifolds 纯流形弦副积的有理模型
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-10-07 DOI: 10.1007/s40062-021-00293-5
Takahito Naito
{"title":"Rational model for the string coproduct of pure manifolds","authors":"Takahito Naito","doi":"10.1007/s40062-021-00293-5","DOIUrl":"10.1007/s40062-021-00293-5","url":null,"abstract":"<div><p>The string coproduct is a coproduct on the homology with field coefficients of the free loop space of a closed oriented manifold introduced by Sullivan in string topology. The coproduct and the Chas-Sullivan loop product give an infinitesimal bialgebra structure on the homology if the Euler characteristic is zero. The aim of this paper is to study the string coproduct using Sullivan models in rational homotopy theory. In particular, we give a rational model for the string coproduct of pure manifolds. Moreover, we study the behavior of the string coproduct in terms of the Hodge decomposition of the rational cohomology of the free loop space. We also give computational examples of the coproduct rationally.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 4","pages":"667 - 702"},"PeriodicalIF":0.5,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4322425","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: 1
On the Maurer-Cartan simplicial set of a complete curved (A_infty )-algebra 关于完全弯曲(A_infty ) -代数的Maurer-Cartan简单集
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-09-25 DOI: 10.1007/s40062-021-00290-8
Niek de Kleijn, Felix Wierstra
{"title":"On the Maurer-Cartan simplicial set of a complete curved (A_infty )-algebra","authors":"Niek de Kleijn,&nbsp;Felix Wierstra","doi":"10.1007/s40062-021-00290-8","DOIUrl":"10.1007/s40062-021-00290-8","url":null,"abstract":"<div><p>In this paper, we develop the <span>(A_infty )</span>-analog of the Maurer-Cartan simplicial set associated to an <span>(L_infty )</span>-algebra and show how we can use this to study the deformation theory of <span>(infty )</span>-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of <span>(A_infty )</span>-algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) <span>(A_infty )</span>-algebras to simplicial sets, which sends a complete curved <span>(A_infty )</span>-algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of <span>(infty )</span>-morphisms of algebras over non-symmetric operads.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 4","pages":"605 - 633"},"PeriodicalIF":0.5,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-021-00290-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4986646","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: 1
Quasi-categories vs. Segal spaces: Cartesian edition 准范畴与西格尔空间:笛卡尔版
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-08-20 DOI: 10.1007/s40062-021-00288-2
Nima Rasekh
{"title":"Quasi-categories vs. Segal spaces: Cartesian edition","authors":"Nima Rasekh","doi":"10.1007/s40062-021-00288-2","DOIUrl":"10.1007/s40062-021-00288-2","url":null,"abstract":"<div><p>We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: </p><ol>\u0000 <li>\u0000 <span>1.</span>\u0000 \u0000 <p>On marked simplicial sets (due to Lurie [31]),</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>2.</span>\u0000 \u0000 <p>On bisimplicial spaces (due to deBrito [12]),</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>3.</span>\u0000 \u0000 <p>On bisimplicial sets,</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>4.</span>\u0000 \u0000 <p>On marked simplicial spaces.</p>\u0000 \u0000 </li>\u0000 </ol><p> The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 4","pages":"563 - 604"},"PeriodicalIF":0.5,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00288-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4780757","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: 7
A cochain level proof of Adem relations in the mod 2 Steenrod algebra mod2 Steenrod代数中Adem关系的协链水平证明
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-08-19 DOI: 10.1007/s40062-021-00287-3
Greg Brumfiel, Anibal Medina-Mardones, John Morgan
{"title":"A cochain level proof of Adem relations in the mod 2 Steenrod algebra","authors":"Greg Brumfiel,&nbsp;Anibal Medina-Mardones,&nbsp;John Morgan","doi":"10.1007/s40062-021-00287-3","DOIUrl":"10.1007/s40062-021-00287-3","url":null,"abstract":"<div><p>In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-<i>i</i> products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod <i>p</i> operations for all primes <i>p</i>. Steenrod’s student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod’s original cochain definition of the Square operations.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 4","pages":"517 - 562"},"PeriodicalIF":0.5,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00287-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5039495","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: 9
Relative singularity categories and singular equivalences 相对奇异范畴和奇异等价
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2021-08-18 DOI: 10.1007/s40062-021-00289-1
Rasool Hafezi
{"title":"Relative singularity categories and singular equivalences","authors":"Rasool Hafezi","doi":"10.1007/s40062-021-00289-1","DOIUrl":"10.1007/s40062-021-00289-1","url":null,"abstract":"<div><p>Let <i>R</i> be a right noetherian ring. We introduce the concept of relative singularity category <span>(Delta _{mathcal {X} }(R))</span> of <i>R</i> with respect to a contravariantly finite subcategory <span>(mathcal {X} )</span> of <span>({text {{mod{-}}}}R.)</span> Along with some finiteness conditions on <span>(mathcal {X} )</span>, we prove that <span>(Delta _{mathcal {X} }(R))</span> is triangle equivalent to a subcategory of the homotopy category <span>(mathbb {K} _mathrm{{ac}}(mathcal {X} ))</span> of exact complexes over <span>(mathcal {X} )</span>. As an application, a new description of the classical singularity category <span>(mathbb {D} _mathrm{{sg}}(R))</span> is given. The relative singularity categories are applied to lift a stable equivalence between two suitable subcategories of the module categories of two given right noetherian rings to get a singular equivalence between the rings. In different types of rings, including path rings, triangular matrix rings, trivial extension rings and tensor rings, we provide some consequences for their singularity categories.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"16 3","pages":"487 - 516"},"PeriodicalIF":0.5,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-021-00289-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4710235","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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