TopologyPub Date : 2007-11-01DOI: 10.1016/j.top.2006.12.004
Ole Andersson
{"title":"A geometric classification of the path components of the space of locally stable maps S3→R4","authors":"Ole Andersson","doi":"10.1016/j.top.2006.12.004","DOIUrl":"10.1016/j.top.2006.12.004","url":null,"abstract":"<div><p>Locally stable maps <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> are classified up to homotopy through locally stable maps. The equivalence class of a map <span><math><mi>f</mi></math></span> is determined by three invariants: the isotopy class <span><math><mi>σ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math></span> of its framed singularity link, the generalized normal degree <span><math><mi>ν</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math></span>, and the algebraic number of cusps <span><math><mi>κ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></math></span> of any extension of <span><math><mi>f</mi></math></span> to a locally stable map of the 4-disk into <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span>. Relations between the invariants are described, and it is proved that for any <span><math><mi>σ</mi></math></span>, <span><math><mi>ν</mi></math></span>, and <span><math><mi>κ</mi></math></span> which satisfy these relations, there exists a map <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> with <span><math><mi>σ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><mi>σ</mi></math></span>, <span><math><mi>ν</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><mi>ν</mi></math></span>, and <span><math><mi>κ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><mi>κ</mi></math></span>. It follows in particular that every framed link in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> is the singularity set of some locally stable map into <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 6","pages":"Pages 577-597"},"PeriodicalIF":0.0,"publicationDate":"2007-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.12.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188212","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}
TopologyPub Date : 2007-11-01DOI: 10.1016/j.top.2007.03.001
Jean-François Lafont , Ivonne J. Ortiz
{"title":"Relative hyperbolicity, classifying spaces, and lower algebraic K-theory","authors":"Jean-François Lafont , Ivonne J. Ortiz","doi":"10.1016/j.top.2007.03.001","DOIUrl":"10.1016/j.top.2007.03.001","url":null,"abstract":"<div><p>For <span><math><mi>Γ</mi></math></span> a relatively hyperbolic group, we construct a model for the universal space among <span><math><mi>Γ</mi></math></span>-spaces with isotropy on the family <span><math><mi>V</mi><mi>C</mi></math></span> of virtually cyclic subgroups of <span><math><mi>Γ</mi></math></span>. We provide a recipe for identifying the maximal infinite virtually cyclic subgroups of Coxeter groups which are lattices in <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mstyle><mi>Isom</mi></mstyle><mrow><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></math></span>. We use the information we obtain to explicitly compute the lower algebraic <span><math><mi>K</mi></math></span>-theory of the Coxeter group <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> (a non-uniform lattice in <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>). Part of this computation involves calculating certain Waldhausen Nil-groups for <span><math><mi>Z</mi><mrow><mo>[</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></mrow></math></span>, <span><math><mi>Z</mi><mrow><mo>[</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>]</mo></mrow></math></span>.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 6","pages":"Pages 527-553"},"PeriodicalIF":0.0,"publicationDate":"2007-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.03.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188329","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}
TopologyPub Date : 2007-11-01DOI: 10.1016/j.top.2007.02.007
Bong H. Lian , Bailin Song
{"title":"Reduced Delzant spaces and a convexity theorem","authors":"Bong H. Lian , Bailin Song","doi":"10.1016/j.top.2007.02.007","DOIUrl":"10.1016/j.top.2007.02.007","url":null,"abstract":"<div><p>The convexity theorem of Atiyah and Guillemin–Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar–Lerman proved that the Marsden–Weinstein reduction of a connected Hamitonian <span><math><mi>G</mi></math></span>-manifold is a stratified symplectic space. Suppose <span><math><mn>1</mn><mo>→</mo><mi>A</mi><mo>→</mo><mi>G</mi><mo>→</mo><mi>T</mi><mo>→</mo><mn>1</mn></math></span> is an exact sequence of compact Lie groups and <span><math><mi>T</mi></math></span> is a torus. Then the reduction of a Hamiltonian <span><math><mi>G</mi></math></span>-manifold with respect to <span><math><mi>A</mi></math></span> yields a Hamiltonian <span><math><mi>T</mi></math></span>-space. We show that if the <span><math><mi>A</mi></math></span>-moment map is proper, then the convexity theorem holds for such a Hamiltonian <span><math><mi>T</mi></math></span>-space, even when it is singular. We also prove that if, furthermore, the <span><math><mi>T</mi></math></span>-space has dimension <span><math><mn>2</mn><mstyle><mi>dim</mi></mstyle><mspace></mspace><mi>T</mi></math></span> and <span><math><mi>T</mi></math></span> acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case. This paper is a concise version of a companion paper [B. Lian. B. Song, A convexity theorem and reduced Delzant spaces, <span>math.DG/0509429</span><svg><path></path></svg>].</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 6","pages":"Pages 554-576"},"PeriodicalIF":0.0,"publicationDate":"2007-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.02.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188311","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}
TopologyPub Date : 2007-09-01DOI: 10.1016/j.top.2007.02.006
Jelena Grbić, Stephen Theriault
{"title":"The homotopy type of the complement of a coordinate subspace arrangement","authors":"Jelena Grbić, Stephen Theriault","doi":"10.1016/j.top.2007.02.006","DOIUrl":"10.1016/j.top.2007.02.006","url":null,"abstract":"<div><p>The homotopy type of the complement of a complex coordinate subspace arrangement is studied by utilising some connections between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 4","pages":"Pages 357-396"},"PeriodicalIF":0.0,"publicationDate":"2007-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.02.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"93385344","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}
TopologyPub Date : 2007-09-01DOI: 10.1016/j.top.2007.03.003
Sangyop Lee
{"title":"Exceptional Dehn fillings on hyperbolic 3-manifolds with at least two boundary components","authors":"Sangyop Lee","doi":"10.1016/j.top.2007.03.003","DOIUrl":"10.1016/j.top.2007.03.003","url":null,"abstract":"<div><p>We estimate the number of exceptional slopes for hyperbolic 3-manifolds with a torus boundary component and at least one other boundary component.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 5","pages":"Pages 437-468"},"PeriodicalIF":0.0,"publicationDate":"2007-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.03.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188355","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}
TopologyPub Date : 2007-09-01DOI: 10.1016/j.top.2007.03.002
Julia E. Bergner
{"title":"Three models for the homotopy theory of homotopy theories","authors":"Julia E. Bergner","doi":"10.1016/j.top.2007.03.002","DOIUrl":"10.1016/j.top.2007.03.002","url":null,"abstract":"<div><p>Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 4","pages":"Pages 397-436"},"PeriodicalIF":0.0,"publicationDate":"2007-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.03.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188342","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}
TopologyPub Date : 2007-09-01DOI: 10.1016/j.top.2007.03.004
M. Kesseböhmer , B.O. Stratmann
{"title":"Homology at infinity; fractal geometry of limiting symbols for modular subgroups","authors":"M. Kesseböhmer , B.O. Stratmann","doi":"10.1016/j.top.2007.03.004","DOIUrl":"10.1016/j.top.2007.03.004","url":null,"abstract":"<div><p>In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive a complete multifractal description of the higher-dimensional level sets arising from the Manin–Marcolli limiting modular symbols.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 5","pages":"Pages 469-491"},"PeriodicalIF":0.0,"publicationDate":"2007-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.03.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188600","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}
TopologyPub Date : 2007-09-01DOI: 10.1016/j.top.2007.01.004
Yanghyun Byun
{"title":"The tangential end fibration of an aspherical Poincaré complex","authors":"Yanghyun Byun","doi":"10.1016/j.top.2007.01.004","DOIUrl":"10.1016/j.top.2007.01.004","url":null,"abstract":"<div><p>We construct a sphere fibration over a finite aspherical Poincaré complex <span><math><mi>X</mi></math></span>, which we call the tangential end fibration, under the condition that the universal cover of <span><math><mi>X</mi></math></span> is forward tame and simply connected at infinity. We show that it is tangent to <span><math><mi>X</mi></math></span> if the formal dimension of <span><math><mi>X</mi></math></span> is even or, when the formal dimension is odd, if the diagonal <span><math><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></math></span> admits a Poincaré embedding structure.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"46 5","pages":"Pages 507-525"},"PeriodicalIF":0.0,"publicationDate":"2007-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.01.004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188266","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}