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The Dold–Thom theorem via factorization homology 因式分解同调的Dold-Thom定理
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-11-10 DOI: 10.1007/s40062-018-0219-1
Lauren Bandklayder
{"title":"The Dold–Thom theorem via factorization homology","authors":"Lauren Bandklayder","doi":"10.1007/s40062-018-0219-1","DOIUrl":"https://doi.org/10.1007/s40062-018-0219-1","url":null,"abstract":"<p>We give a new proof of the classical Dold–Thom theorem using factorization homology. Our method is direct and conceptual, avoiding the Eilenberg–Steenrod axioms entirely in favor of a more general geometric argument.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"579 - 593"},"PeriodicalIF":0.5,"publicationDate":"2018-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0219-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4430487","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
The Koszul–Tate type resolution for Gerstenhaber–Batalin–Vilkovisky algebras Gerstenhaber-Batalin-Vilkovisky代数的Koszul-Tate型解析
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-10-25 DOI: 10.1007/s40062-018-0218-2
Jeehoon Park, Donggeon Yhee
{"title":"The Koszul–Tate type resolution for Gerstenhaber–Batalin–Vilkovisky algebras","authors":"Jeehoon Park,&nbsp;Donggeon Yhee","doi":"10.1007/s40062-018-0218-2","DOIUrl":"https://doi.org/10.1007/s40062-018-0218-2","url":null,"abstract":"<p>Tate provided an <i>explicit</i> way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring <i>R</i> in Tate (Ill J Math 1:14–27, 1957). The goal of this article is to generalize his result to the case of GBV (Gerstenhaber–Batalin–Vilkovisky) algebras and, more generally, the descendant <span>(L_infty )</span>-algebras. More precisely, for a given GBV algebra <span>((mathcal {A}=oplus _{mge 0}mathcal {A}_m, delta , ell _2^delta ))</span>, we provide another <i>explicit</i> GBV algebra <span>((widetilde{mathcal {A}}=oplus _{mge 0}widetilde{mathcal {A}}_m, widetilde{delta }, ell _2^{widetilde{delta }}))</span> such that its total homology is the same as the degree zero part of the homology <span>(H_0(mathcal {A}, delta ))</span> of the given GBV algebra <span>((mathcal {A}, delta , ell _2^delta ))</span>.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"455 - 475"},"PeriodicalIF":0.5,"publicationDate":"2018-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0218-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4985810","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 combinatorial model for the path fibration 路径颤振的组合模型
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-09-29 DOI: 10.1007/s40062-018-0216-4
Manuel Rivera, Samson Saneblidze
{"title":"A combinatorial model for the path fibration","authors":"Manuel Rivera,&nbsp;Samson Saneblidze","doi":"10.1007/s40062-018-0216-4","DOIUrl":"https://doi.org/10.1007/s40062-018-0216-4","url":null,"abstract":"<p>We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set <i>X</i> we associate a necklical set <span>({widehat{{varvec{Omega }}}}X)</span> such that its geometric realization <span>(|{widehat{{varvec{Omega }}}}X|)</span>, a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on |<i>X</i>| and the differential graded module of chains <span>(C_*({widehat{{varvec{Omega }}}}X))</span> is a differential graded associative algebra generalizing Adams’ cobar construction.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"393 - 410"},"PeriodicalIF":0.5,"publicationDate":"2018-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0216-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5133041","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
Topology of scrambled simplices 乱序单形的拓扑结构
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-09-27 DOI: 10.1007/s40062-018-0214-6
Dmitry N. Kozlov
{"title":"Topology of scrambled simplices","authors":"Dmitry N. Kozlov","doi":"10.1007/s40062-018-0214-6","DOIUrl":"https://doi.org/10.1007/s40062-018-0214-6","url":null,"abstract":"<p>In this paper we define a?family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a?standard <i>d</i>-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a?<span>(Delta )</span>-complex. As our main result, we completely determine the homotopy type of these spaces. In fact, somewhat surprisingly, we are able to prove that each of them is either contractible or homotopy equivalent to an?odd-dimensional sphere. We develop the language to determine the homotopy type directly from the combinatorics of the indexing word. As added benefit of our investigation, we are able to emulate the Dunce hat phenomenon, and to obtain a?large family of both <span>(Delta )</span>-complexes, as well as simplicial complexes, which are contractible, but not collapsible.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"371 - 391"},"PeriodicalIF":0.5,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0214-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5064663","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: 3
Unstable splittings in Hodge filtered Brown–Peterson cohomology Hodge滤波Brown-Peterson上同中的不稳定分裂
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-09-27 DOI: 10.1007/s40062-018-0215-5
Gereon Quick
{"title":"Unstable splittings in Hodge filtered Brown–Peterson cohomology","authors":"Gereon Quick","doi":"10.1007/s40062-018-0215-5","DOIUrl":"https://doi.org/10.1007/s40062-018-0215-5","url":null,"abstract":"<p>We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown–Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson’s unstable splitting. As a consequence, we obtain an analog of Quillen’s theorem for Hodge filtered Brown–Peterson cohomology for complex manifolds.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"349 - 369"},"PeriodicalIF":0.5,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0215-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5066595","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 topological computation of (K_4) of the Gaussian and Eisenstein integers 高斯整数和爱森斯坦整数(K_4)的拓扑计算
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-08-18 DOI: 10.1007/s40062-018-0212-8
Mathieu Dutour Sikirić, Herbert Gangl, Paul E. Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki
{"title":"On the topological computation of (K_4) of the Gaussian and Eisenstein integers","authors":"Mathieu Dutour Sikirić,&nbsp;Herbert Gangl,&nbsp;Paul E. Gunnells,&nbsp;Jonathan Hanke,&nbsp;Achill Schürmann,&nbsp;Dan Yasaki","doi":"10.1007/s40062-018-0212-8","DOIUrl":"https://doi.org/10.1007/s40062-018-0212-8","url":null,"abstract":"<p>In this paper we use topological tools to investigate the structure of the algebraic <i>K</i>-groups <span>(K_4(R))</span> for <span>(R=Z[i])</span> and <span>(R=Z[rho ])</span> where <span>(i := sqrt{-1})</span> and <span>(rho := (1+sqrt{-3})/2)</span>. We exploit the close connection between homology groups of <span>(mathrm {GL}_n(R))</span> for <span>(nle 5)</span> and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which <span>(mathrm {GL}_n(R))</span> acts. Our main result is that <span>(K_{4} ({mathbb {Z}}[i]))</span> and <span>(K_{4} ({mathbb {Z}}[rho ]))</span> have no <i>p</i>-torsion for <span>(pge 5)</span>.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"281 - 291"},"PeriodicalIF":0.5,"publicationDate":"2018-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0212-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4709713","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
Equivariant formality of isotropic torus actions 各向同性环面作用的等变形式
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-07-24 DOI: 10.1007/s40062-018-0207-5
Jeffrey D. Carlson
{"title":"Equivariant formality of isotropic torus actions","authors":"Jeffrey D. Carlson","doi":"10.1007/s40062-018-0207-5","DOIUrl":"https://doi.org/10.1007/s40062-018-0207-5","url":null,"abstract":"<p>Considering the potential equivariant formality of the left action of a connected Lie group <i>K</i> on the homogeneous space <i>G</i>?/?<i>K</i>, we arrive through a sequence of reductions at the case <i>G</i> is compact and simply-connected and <i>K</i> is a torus. We then classify all pairs (<i>G</i>,?<i>S</i>) such that <i>G</i> is compact connected Lie and the embedded circular subgroup <i>S</i> acts equivariantly formally on <i>G</i>?/?<i>S</i>. In the process we provide what seems to be the first published proof of the structure (known to Leray and Koszul) of the cohomology rings</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 1","pages":"199 - 234"},"PeriodicalIF":0.5,"publicationDate":"2018-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0207-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4937331","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
Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebras with infinite dimensional coefficients 无穷维系数的cones - moscovici Hopf代数的Hopf-循环上同调
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-04-28 DOI: 10.1007/s40062-018-0205-7
B. Rangipour, S. Sütlü, F. Yazdani Aliabadi
{"title":"Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebras with infinite dimensional coefficients","authors":"B. Rangipour,&nbsp;S. Sütlü,&nbsp;F. Yazdani Aliabadi","doi":"10.1007/s40062-018-0205-7","DOIUrl":"https://doi.org/10.1007/s40062-018-0205-7","url":null,"abstract":"<p>We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebra <span>(mathcal{H}_n)</span>. More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of <span>(mathcal{H}_n)</span>, and we show that the van Est type characteristic homomorphism from the Hopf-cyclic complex of <span>(mathcal{H}_n)</span> to the Gelfand–Fuks cohomology of the Lie algebra <span>(W_n)</span> of formal vector fields on <span>({mathbb {R}}^n)</span> respects this multiplicative structure. We then illustrate the machinery for <span>(n=1)</span>.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"13 4","pages":"927 - 969"},"PeriodicalIF":0.5,"publicationDate":"2018-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0205-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5071478","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
A simplicial foundation for differential and sector forms in tangent categories 切线范畴中微分形式和扇形形式的一个简单基础
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-04-26 DOI: 10.1007/s40062-018-0204-8
G. S. H. Cruttwell, Rory B. B. Lucyshyn-Wright
{"title":"A simplicial foundation for differential and sector forms in tangent categories","authors":"G. S. H. Cruttwell,&nbsp;Rory B. B. Lucyshyn-Wright","doi":"10.1007/s40062-018-0204-8","DOIUrl":"https://doi.org/10.1007/s40062-018-0204-8","url":null,"abstract":"<p>Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections. In this paper we investigate differential and <i>sector</i> forms in tangent categories. We show that sector forms in any tangent category have a rich structure: they form a symmetric cosimplicial object. This appears to be a new result in differential geometry, even for smooth manifolds. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de Rham complex of differential forms, which may be identified with <i>alternating</i> sector forms. Further, the symmetric cosimplicial structure on sector forms arises naturally through a new equational presentation of symmetric cosimplicial objects, which we develop herein.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"13 4","pages":"867 - 925"},"PeriodicalIF":0.5,"publicationDate":"2018-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0204-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5386542","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
Gorenstein AC-projective complexes 戈伦斯坦ac投影复合体
IF 0.5 4区 数学
Journal of Homotopy and Related Structures Pub Date : 2018-03-20 DOI: 10.1007/s40062-018-0203-9
James Gillespie
{"title":"Gorenstein AC-projective complexes","authors":"James Gillespie","doi":"10.1007/s40062-018-0203-9","DOIUrl":"https://doi.org/10.1007/s40062-018-0203-9","url":null,"abstract":"<p>Let <i>R</i> be any ring with identity and <span>( Ch (R))</span> the category of chain complexes of (left) <i>R</i>-modules. We show that the Gorenstein AC-projective chain complexes of?[1] are the cofibrant objects of an abelian model structure on <span>( Ch (R))</span>. The model structure is cofibrantly generated and is projective in the sense that the trivially cofibrant objects are the categorically projective chain complexes. We show that when <i>R</i> is a Ding-Chen ring, that is, a two-sided coherent ring with finite self FP-injective dimension, then the model structure is finitely generated, and so its homotopy category is compactly generated. Constructing this model structure also shows that every chain complex over any ring has a Gorenstein AC-projective precover. These are precisely Gorenstein projective (in the usual sense) precovers whenever <i>R</i> is either a Ding-Chen ring, or, a ring for which all level (left) <i>R</i>-modules have finite projective dimension. For a general (right) coherent ring <i>R</i>, the Gorenstein AC-projective complexes coincide with the Ding projective complexes of [31] and so provide such precovers in this case.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"13 4","pages":"769 - 791"},"PeriodicalIF":0.5,"publicationDate":"2018-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0203-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5095229","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
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