Topology最新文献

{"title":"A descent principle for the Dirac–dual-Dirac method","authors":"Heath Emerson ,&nbsp;Ralf Meyer","doi":"10.1016/j.top.2007.02.001","DOIUrl":"10.1016/j.top.2007.02.001","url":null,"abstract":"<div><p>Let <span><math><mi>G</mi></math></span> be a torsion-free discrete group with a finite-dimensional classifying space <span><math><mi>B</mi><mi>G</mi></math></span>. We show that <span><math><mi>G</mi></math></span> has a dual-Dirac morphism if and only if a certain coarse (co-)assembly map is an isomorphism. Hence the existence of a dual-Dirac morphism for such groups is a metric, that is, coarse, invariant. We get results for groups with torsion as well.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 2","pages":"Pages 185-209"},"PeriodicalIF":0.0,"publicationDate":"2007-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.02.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Information for Authors","authors":"","doi":"10.1016/S0040-9383(07)00019-5","DOIUrl":"https://doi.org/10.1016/S0040-9383(07)00019-5","url":null,"abstract":"","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 2","pages":"Page IFC"},"PeriodicalIF":0.0,"publicationDate":"2007-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0040-9383(07)00019-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137255536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Boundaries of cocompact proper CAT(0) spaces","authors":"Ross Geoghegan,&nbsp;Pedro Ontaneda","doi":"10.1016/j.top.2006.12.002","DOIUrl":"10.1016/j.top.2006.12.002","url":null,"abstract":"<div><p>A proper CAT(0) metric space <span><math><mi>X</mi></math></span> is <em>cocompact</em> if it has a compact generating domain with respect to its full isometry group. Any proper CAT(0) space, cocompact or not, has a compact metrizable boundary at infinity <span><math><msub><mrow><mi>∂</mi></mrow><mrow><mi>∞</mi></mrow></msub><mi>X</mi></math></span>; indeed, up to homeomorphism, this boundary is arbitrary. However, cocompactness imposes restrictions on what the boundary can be. Swenson showed that the boundary of a cocompact <span><math><mi>X</mi></math></span> has to be finite-dimensional. Here we show more: the dimension of <span><math><msub><mrow><mi>∂</mi></mrow><mrow><mi>∞</mi></mrow></msub><mi>X</mi></math></span> has to be equal to the global Čech cohomological dimension of <span><math><msub><mrow><mi>∂</mi></mrow><mrow><mi>∞</mi></mrow></msub><mi>X</mi></math></span>. For example: a compact manifold with non-empty boundary cannot be <span><math><msub><mrow><mi>∂</mi></mrow><mrow><mi>∞</mi></mrow></msub><mi>X</mi></math></span> with <span><math><mi>X</mi></math></span> cocompact. We include two consequences of this topological/geometric fact: (1) The dimension of the boundary is a quasi-isometry invariant of CAT(0) groups. (2) Geodesic segments in a cocompact <span><math><mi>X</mi></math></span> can “almost” be extended to geodesic rays, i.e. <span><math><mi>X</mi></math></span> is almost geodesically complete.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 2","pages":"Pages 129-137"},"PeriodicalIF":0.0,"publicationDate":"2007-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.12.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A solution to de Groot’s absolute cone conjecture","authors":"Craig R. Guilbault","doi":"10.1016/j.top.2006.12.001","DOIUrl":"10.1016/j.top.2006.12.001","url":null,"abstract":"<div><p>A compactum <span><math><mi>X</mi></math></span> is an ‘absolute cone’ if, for each of its points <span><math><mi>x</mi></math></span>, the space <span><math><mi>X</mi></math></span> is homeomorphic to a cone with <span><math><mi>x</mi></math></span> corresponding to the cone point. In 1971, J. de Groot conjectured that each <span><math><mi>n</mi></math></span>-dimensional absolute cone is an <span><math><mi>n</mi></math></span>-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for <span><math><mi>n</mi><mo>≤</mo><mn>3</mn></math></span> and false for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. For <span><math><mi>n</mi><mo>=</mo><mn>4</mn></math></span>, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 2","pages":"Pages 89-102"},"PeriodicalIF":0.0,"publicationDate":"2007-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.12.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Betti numbers of finitely presented groups and very rapidly growing functions","authors":"Alexander Nabutovsky ,&nbsp;Shmuel Weinberger","doi":"10.1016/j.top.2007.02.002","DOIUrl":"10.1016/j.top.2007.02.002","url":null,"abstract":"<div><p>Define the length of a finite presentation of a group <span><math><mi>G</mi></math></span> as the sum of lengths of all relators plus the number of generators. How large can the <span><math><mi>k</mi></math></span>th Betti number <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo></math></span> rank <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span> be providing that <span><math><mi>G</mi></math></span> has length <span><math><mo>≤</mo><mi>N</mi></math></span> and <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span> is finite? We prove that for every <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> the maximum <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></math></span> of the <span><math><mi>k</mi></math></span>th Betti numbers of all such groups is an extremely rapidly growing function of <span><math><mi>N</mi></math></span>. It grows faster that all functions previously encountered in mathematics (outside of logic) including non-computable functions (at least those that are known to us). More formally, <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> grows as the third busy beaver function that measures the maximal productivity of Turing machines with <span><math><mo>≤</mo><mi>N</mi></math></span> states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines.</p><p>We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function.</p><p>Also, we outline a construction of a finitely presented group all of whose homology groups are either <span><math><mstyle><mi>Z</mi></mstyle></math></span> or trivial such that its Betti numbers form a random binary sequence.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 2","pages":"Pages 211-223"},"PeriodicalIF":0.0,"publicationDate":"2007-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.02.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Little cubes and long knots","authors":"Ryan Budney","doi":"10.1016/j.top.2006.09.001","DOIUrl":"10.1016/j.top.2006.09.001","url":null,"abstract":"<div><p>This paper gives a partial description of the homotopy type of <span><math><mi>K</mi></math></span>, the space of long knots in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. The primary result is the construction of a homotopy equivalence <span><math><mi>K</mi><mo>≃</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>P</mi><mo>⊔</mo><mrow><mo>{</mo><mo>∗</mo><mo>}</mo></mrow><mo>)</mo></mrow></math></span> where <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>P</mi><mo>⊔</mo><mrow><mo>{</mo><mo>∗</mo><mo>}</mo></mrow><mo>)</mo></mrow></math></span> is the free little 2-cubes object on the pointed space <span><math><mi>P</mi><mo>⊔</mo><mrow><mo>{</mo><mo>∗</mo><mo>}</mo></mrow></math></span>, where <span><math><mi>P</mi><mo>⊂</mo><mi>K</mi></math></span> is the subspace of prime knots, and <span><math><mo>∗</mo></math></span> is a disjoint base-point. In proving the freeness result, a close correspondence is discovered between the Jaco–Shalen–Johannson decomposition of knot complements and the little cubes action on <span><math><mi>K</mi></math></span>. Beyond studying long knots in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> we show that for any compact manifold <span><math><mi>M</mi></math></span> the space of embeddings of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><mi>M</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><mi>M</mi></math></span> with support in <span><math><msup><mrow><mstyle><mi>I</mi></mstyle></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><mi>M</mi></math></span> admits an action of the operad of little <span><math><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-cubes. If <span><math><mi>M</mi><mo>=</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> this embedding space is the space of framed long <span><math><mi>n</mi></math></span>-knots in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup></math></span>, and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 1","pages":"Pages 1-27"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.09.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Rasmussen invariants and the sharper slice-Bennequin inequality on knots","authors":"Tomomi Kawamura","doi":"10.1016/j.top.2006.10.001","DOIUrl":"10.1016/j.top.2006.10.001","url":null,"abstract":"<div><p>Rasmussen introduced a knot invariant based on Khovanov homology theory, and showed that this invariant estimates the four-genus of knots. We compare his result with the sharper slice-Bennequin inequality for knots. Then we obtain a similar estimate of the Rasmussen invariant for this inequality.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 1","pages":"Pages 29-38"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.10.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Information for Authors","authors":"","doi":"10.1016/S0040-9383(06)00058-9","DOIUrl":"https://doi.org/10.1016/S0040-9383(06)00058-9","url":null,"abstract":"","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 1","pages":"Page IFC"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0040-9383(06)00058-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91631534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On spineless cacti, Deligne’s conjecture and Connes–Kreimer’s Hopf algebra","authors":"Ralph M. Kaufmann","doi":"10.1016/j.top.2006.10.002","DOIUrl":"10.1016/j.top.2006.10.002","url":null,"abstract":"<div><p>Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne’s conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 1","pages":"Pages 39-88"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.10.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Volume Contents and Author Index","authors":"","doi":"10.1016/S0040-9383(06)00052-8","DOIUrl":"https://doi.org/10.1016/S0040-9383(06)00052-8","url":null,"abstract":"","PeriodicalId":54424,"journal":{"name":"Topology","volume":"45 6","pages":"Pages III-VIII"},"PeriodicalIF":0.0,"publicationDate":"2006-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0040-9383(06)00052-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137125788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}