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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
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
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