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Journal of Homotopy and Related Structures最新文献

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Unitary calculus: model categories and convergence 一元微积分:模型范畴与收敛性
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-08-09 DOI: 10.1007/s40062-022-00311-0
Niall Taggart
{"title":"Unitary calculus: model categories and convergence","authors":"Niall Taggart","doi":"10.1007/s40062-022-00311-0","DOIUrl":"10.1007/s40062-022-00311-0","url":null,"abstract":"<div><p>We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra—a variation of orthogonal spectra—as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the <i>n</i>-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 3","pages":"419 - 462"},"PeriodicalIF":0.5,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-022-00311-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4371864","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: 7
Modeling bundle-valued forms on the path space with a curved iterated integral 用曲线迭代积分在路径空间上对束值形式进行建模
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-07-13 DOI: 10.1007/s40062-022-00306-x
Cheyne Glass, Corbett Redden
{"title":"Modeling bundle-valued forms on the path space with a curved iterated integral","authors":"Cheyne Glass,&nbsp;Corbett Redden","doi":"10.1007/s40062-022-00306-x","DOIUrl":"10.1007/s40062-022-00306-x","url":null,"abstract":"<div><p>The usual iterated integral map given by Chen produces an equivalence between the two-sided bar complex on differential forms and the de Rham complex on the path space. This map fails to make sense when considering the curved differential graded algebra of bundle-valued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen’s iterated integral that incorporates parallel transport and maps an analog of the two-sided bar construction on bundle-valued forms to bundle-valued forms on the path space. This iterated integral is proven to be a homotopy equivalence of curved differential graded algebras, and for real-valued forms it factors through the usual Chen iterated integral.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 3","pages":"309 - 353"},"PeriodicalIF":0.5,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-022-00306-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4540337","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
Correction to: A cochain level proof of Adem relations in the mod 2 Steenrod algebra 修正:mod2 Steenrod代数中Adem关系的一个协链水平证明
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-07-04 DOI: 10.1007/s40062-022-00307-w
Greg Brumfiel, Anibal Medina-Mardones, John Morgan
{"title":"Correction to: 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/s40062-022-00307-w","DOIUrl":"10.1007/s40062-022-00307-w","url":null,"abstract":"","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 3","pages":"463 - 463"},"PeriodicalIF":0.5,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-022-00307-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4167867","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
Cohomology and deformations of twisted Rota–Baxter operators and NS-algebras 扭曲Rota-Baxter算子和ns -代数的上同调和变形
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-05-05 DOI: 10.1007/s40062-022-00305-y
Apurba Das
{"title":"Cohomology and deformations of twisted Rota–Baxter operators and NS-algebras","authors":"Apurba Das","doi":"10.1007/s40062-022-00305-y","DOIUrl":"10.1007/s40062-022-00305-y","url":null,"abstract":"<div><p>The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an <span>(L_infty )</span>-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This leads to cohomology associated to a twisted Rota–Baxter operator. This cohomology can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota–Baxter operators by means of the above-defined cohomology. Application is given to Reynolds operators. In the second part, we consider NS-algebras of Leroux that are related to twisted Rota–Baxter operators in the same way dendriform algebras are related to Rota–Baxter operators. We define cohomology of NS-algebras using non-symmetric operads and study their deformations in terms of the cohomology.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 2","pages":"233 - 262"},"PeriodicalIF":0.5,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4221435","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: 12
On Lusternik–Schnirelmann category and topological complexity of non-k-equal manifolds 非k相等流形的Lusternik-Schnirelmann范畴和拓扑复杂度
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-04-25 DOI: 10.1007/s40062-022-00304-z
Jesús González, José Luis León-Medina
{"title":"On Lusternik–Schnirelmann category and topological complexity of non-k-equal manifolds","authors":"Jesús González,&nbsp;José Luis León-Medina","doi":"10.1007/s40062-022-00304-z","DOIUrl":"10.1007/s40062-022-00304-z","url":null,"abstract":"<div><p>We compute the Lusternik–Schnirelmann category and all the higher topological complexities of non-<i>k</i>-equal manifolds <span>(M_d^{(k)}(n))</span> for certain values of <i>d</i>, <i>k</i> and <i>n</i>. This includes instances where <span>(M_d^{(k)}(n))</span> is known to be rationally non-formal. The key ingredient in our computations is the knowledge of the cohomology ring <span>(H^*(M_d^{(k)}(n)))</span> as described by Dobrinskaya and Turchin. A fine tuning comes from the use of obstruction theory techniques.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 2","pages":"217 - 231"},"PeriodicalIF":0.5,"publicationDate":"2022-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4957836","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
({ mathsf {TQ} })-completion and the Taylor tower of the identity functor ({ mathsf {TQ} })-补全和恒等函子的泰勒塔
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-03-30 DOI: 10.1007/s40062-022-00303-0
Nikolas Schonsheck
{"title":"({ mathsf {TQ} })-completion and the Taylor tower of the identity functor","authors":"Nikolas Schonsheck","doi":"10.1007/s40062-022-00303-0","DOIUrl":"10.1007/s40062-022-00303-0","url":null,"abstract":"<div><p>The goal of this short paper is to study the convergence of the Taylor tower of the identity functor in the context of operadic algebras in spectra. Specifically, we show that if <i>A</i> is a <span>((-1))</span>-connected <span>({ mathcal {O} })</span>-algebra with 0-connected <span>({ mathsf {TQ} })</span>-homology spectrum <span>({ mathsf {TQ} }(A))</span>, then there is a natural weak equivalence <span>(P_infty ({ mathrm {id} })A simeq A^wedge _{ mathsf {TQ} })</span> between the limit of the Taylor tower of the identity functor evaluated on <i>A</i> and the <span>({ mathsf {TQ} })</span>-completion of <i>A</i>. Since, in this context, the identity functor is only known to be 0-analytic, this result extends knowledge of the Taylor tower of the identity beyond its “radius of convergence.”</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 2","pages":"201 - 216"},"PeriodicalIF":0.5,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5156569","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: 2
Resolutions of operads via Koszul (bi)algebras 通过Koszul (bi)代数解析操作数
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-03-03 DOI: 10.1007/s40062-022-00302-1
Pedro Tamaroff
{"title":"Resolutions of operads via Koszul (bi)algebras","authors":"Pedro Tamaroff","doi":"10.1007/s40062-022-00302-1","DOIUrl":"10.1007/s40062-022-00302-1","url":null,"abstract":"<div><p>We introduce a construction that produces from each bialgebra <i>H</i> an operad <span>(mathsf {Ass}_H)</span> controlling associative algebras in the monoidal category of <i>H</i>-modules or, briefly, <i>H</i>-algebras. When the underlying algebra of this bialgebra is Koszul, we give explicit formulas for the minimal model of this operad depending only on the coproduct of <i>H</i> and the Koszul model of <i>H</i>. This operad is seldom quadratic—and hence does not fall within the reach of Koszul duality theory—so our work provides a new rich family of examples where an explicit minimal model of an operad can be obtained. As an application, we observe that if we take <i>H</i> to be the mod-2 Steenrod algebra <span>({mathscr {A}})</span>, then this notion of an associative <i>H</i>-algebra coincides with the usual notion of an <span>(mathscr {A})</span>-algebra considered by homotopy theorists. This makes available to us an operad <span>(mathsf {Ass}_{{mathscr {A}}})</span> along with its minimal model that controls the category of associative <span>({mathscr {A}})</span>-algebras, and the notion of strong homotopy associative <span>({mathscr {A}})</span>-algebras.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 2","pages":"175 - 200"},"PeriodicalIF":0.5,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-022-00302-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4131122","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 Euler–Poincaré characteristics of a simply connected rationally elliptic CW-complex 单连通合理椭圆型cw -复形的euler - poincarcarr特征
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-02-22 DOI: 10.1007/s40062-022-00301-2
Mahmoud Benkhalifa
{"title":"On the Euler–Poincaré characteristics of a simply connected rationally elliptic CW-complex","authors":"Mahmoud Benkhalifa","doi":"10.1007/s40062-022-00301-2","DOIUrl":"10.1007/s40062-022-00301-2","url":null,"abstract":"<div><p>For a simply connected rationally elliptic CW-complex <i>X</i>, we show that the cohomology and the homotopy Euler–Poincaré characteristics are related to two new numerical invariants namely <span>(eta _{X})</span> and <span>(rho _{X})</span> which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of <i>X</i>.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 2","pages":"163 - 174"},"PeriodicalIF":0.5,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4851561","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
Connectedness of graphs arising from the dual Steenrod algebra 对偶Steenrod代数图的连通性
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-02-08 DOI: 10.1007/s40062-022-00300-3
Donald M. Larson
{"title":"Connectedness of graphs arising from the dual Steenrod algebra","authors":"Donald M. Larson","doi":"10.1007/s40062-022-00300-3","DOIUrl":"10.1007/s40062-022-00300-3","url":null,"abstract":"<div><p>We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra <span>(mathscr {A}^*)</span>. We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of <span>(mathscr {A}^*)</span> and its structure as a Hopf algebra.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 1","pages":"145 - 161"},"PeriodicalIF":0.5,"publicationDate":"2022-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4337751","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 graded ({mathbb {E}}_{infty })-rings and projective schemes in spectral algebraic geometry 谱代数几何中的分级({mathbb {E}}_{infty }) -环和射影格式
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2022-01-31 DOI: 10.1007/s40062-021-00298-0
Mariko Ohara, Takeshi Torii
{"title":"On graded ({mathbb {E}}_{infty })-rings and projective schemes in spectral algebraic geometry","authors":"Mariko Ohara,&nbsp;Takeshi Torii","doi":"10.1007/s40062-021-00298-0","DOIUrl":"10.1007/s40062-021-00298-0","url":null,"abstract":"<div><p>We introduce graded <span>({mathbb {E}}_{infty })</span>-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective <span>({mathbb {N}})</span>-graded <span>({mathbb {E}}_{infty })</span>-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the <span>(infty )</span>-category of almost perfect quasi-coherent sheaves over a spectral projective scheme <span>(text { {Proj}},(A))</span> associated to a connective <span>({mathbb {N}})</span>-graded <span>({mathbb {E}}_{infty })</span>-ring <i>A</i> can be described in terms of <span>({{mathbb {Z}}})</span>-graded <i>A</i>-modules.</p></div>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"17 1","pages":"105 - 144"},"PeriodicalIF":0.5,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40062-021-00298-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5179992","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
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