Expositiones Mathematicae最新文献

{"title":"Relative plus constructions","authors":"Guille Carrión Santiago ,&nbsp;Jérôme Scherer","doi":"10.1016/j.exmath.2023.03.001","DOIUrl":"10.1016/j.exmath.2023.03.001","url":null,"abstract":"<div><p>Let <span><math><mi>h</mi></math></span> be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></math></span>, consisting of a connected space <span><math><mi>X</mi></math></span> and an <span><math><mi>h</mi></math></span>-perfect normal subgroup <span><math><mi>H</mi></math></span> of the fundamental group <span><math><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span>, an <span><math><mi>h</mi></math></span>-acyclic map <span><math><mrow><mi>X</mi><mo>→</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>H</mi></mrow><mrow><mo>+</mo><mi>h</mi></mrow></msubsup></mrow></math></span> inducing the quotient by <span><math><mi>H</mi></math></span> on the fundamental group. We show that this map is terminal among the <span><math><mi>h</mi></math></span>-acyclic maps that kill a subgroup of <span><math><mi>H</mi></math></span>. When <span><math><mi>h</mi></math></span> is an ordinary homology theory with coefficients in a commutative ring with unit <span><math><mi>R</mi></math></span>, this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen’s plus construction. We also clarify the necessity to use a strongly <span><math><mi>R</mi></math></span>-perfect group <span><math><mi>H</mi></math></span> in characteristic zero.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41568602","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}

{"title":"A unified view on the functorial nerve theorem and its variations","authors":"Ulrich Bauer ,&nbsp;Michael Kerber ,&nbsp;Fabian Roll ,&nbsp;Alexander Rolle","doi":"10.1016/j.exmath.2023.04.005","DOIUrl":"10.1016/j.exmath.2023.04.005","url":null,"abstract":"<div><p><span>The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by </span>closed convex sets<span><span><span> in Euclidean space, and for covers of a </span>simplicial complex by </span>subcomplexes<span>. Then, we establish a more general, “unified” nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory.</span></span></p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48967918","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}

{"title":"Geometric quadratic Chabauty and p-adic heights","authors":"Juanita Duque-Rosero, Sachi Hashimoto, P. Spelier","doi":"10.1016/j.exmath.2023.05.003","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.05.003","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47399306","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}

{"title":"Degeneration locus of Qp-local systems: Conjectures","authors":"A. Cadoret","doi":"10.1016/j.exmath.2023.05.002","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.05.002","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44357718","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}

{"title":"Coset topologies on Z and arithmetic applications","authors":"Ignazio Longhi,&nbsp;Yunzhu Mu ,&nbsp;Francesco Maria Saettone","doi":"10.1016/j.exmath.2022.10.001","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.10.001","url":null,"abstract":"<div><p>We provide a construction which covers as special cases many of the topologies on integers one can find in the literature. Moreover, our analysis of the Golomb and Kirch topologies inserts them in a family of connected, Hausdorff topologies on <span><math><mi>Z</mi></math></span>, obtained from closed sets of the profinite completion <span><math><mover><mrow><mi>Z</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. We also discuss various applications to number theory.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49834270","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}

{"title":"On Stiefel’s parallelizability of 3-manifolds","authors":"Valentina Bais ,&nbsp;Daniele Zuddas","doi":"10.1016/j.exmath.2023.01.001","DOIUrl":"10.1016/j.exmath.2023.01.001","url":null,"abstract":"<div><p>We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42914149","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}

{"title":"Groups of prime degree and the Bateman–Horn Conjecture","authors":"Gareth A. Jones ,&nbsp;Alexander K. Zvonkin","doi":"10.1016/j.exmath.2022.11.002","DOIUrl":"10.1016/j.exmath.2022.11.002","url":null,"abstract":"<div><p>As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree <span><math><mrow><mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> is prime. We present heuristic arguments and computational evidence based on the Bateman–Horn Conjecture to support a conjecture that for each prime <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span> there are infinitely many primes of this form, even if one restricts to prime values of <span><math><mi>q</mi></math></span>. Similar arguments and results apply to the parameters of the simple groups <span><math><mrow><msub><mrow><mi>PSL</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>PSU</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>PSp</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> which arise in the work of Dixon and Zalesskii on linear groups of prime degree.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43034572","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}

{"title":"Rings of tautological forms on moduli spaces of curves","authors":"Robin de Jong, Stefan van der Lugt","doi":"10.1016/j.exmath.2023.02.008","DOIUrl":"https://doi.org/10.1016/j.exmath.2023.02.008","url":null,"abstract":"","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"54342561","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}

{"title":"Krull-Remak-Schmidt decompositions in Hom-finite additive categories","authors":"Amit Shah","doi":"10.1016/j.exmath.2022.12.003","DOIUrl":"10.1016/j.exmath.2022.12.003","url":null,"abstract":"<div><p>An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, a finite direct sum decomposition consisting of objects with local endomorphism rings—is known as a Krull-Schmidt category. A <span><math><mo>Hom</mo></math></span>-finite category is an additive category <span><math><mi>A</mi></math></span> for which there is a commutative unital ring <span><math><mi>k</mi></math></span>, such that each <span><math><mo>Hom</mo></math></span>-set in <span><math><mi>A</mi></math></span> is a finite length <span><math><mi>k</mi></math></span>-module. The aim of this note is to provide a proof that a <span><math><mo>Hom</mo></math></span>-finite category is Krull-Schmidt, if and only if it has split idempotents, if and only if each indecomposable object has a local endomorphism ring.</p></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41496827","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}

{"title":"Noncommutative Ck functions and Fréchet derivatives of operator functions","authors":"Evangelos A. Nikitopoulos","doi":"10.1016/j.exmath.2022.12.004","DOIUrl":"https://doi.org/10.1016/j.exmath.2022.12.004","url":null,"abstract":"<div><p>Fix a unital <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-algebra <span><math><mi>A</mi></math></span>, and write <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub></math></span> for the set of self-adjoint elements of <span><math><mi>A</mi></math></span>. Also, if <span><math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>ℂ</mi></mrow></math></span> is a continuous function, then write <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>→</mo><mi>A</mi></mrow></math></span> for the <em>operator function</em> <span><math><mrow><mi>a</mi><mo>↦</mo><mi>f</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span> defined via functional calculus. In this paper, we introduce and study a space <span><math><mrow><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> functions <span><math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>ℂ</mi></mrow></math></span> such that, no matter the choice of <span><math><mi>A</mi></math></span>, the operator function <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>→</mo><mi>A</mi></mrow></math></span> is <span><math><mi>k</mi></math></span>-times continuously Fréchet differentiable. In other words, if <span><math><mrow><mi>f</mi><mo>∈</mo><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, then <span><math><mi>f</mi></math></span> “lifts” to a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> map <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>:</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>→</mo><mi>A</mi></mrow></math></span>, for any (possibly noncommutative) unital <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-algebra <span><math><mi>A</mi></math></span>. For this reason, we call <span><math><mrow><mi>N</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> the space of <em>noncommutative</em> <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> <em>functions</em>. Our proof that <span><math><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>sa</mi></mrow></msub><mo>;</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span>, which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estim","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49834269","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}