Topology最新文献

{"title":"The behavior of Nil-groups under localization and the relative assembly map","authors":"Joachim Grunewald","doi":"10.1016/j.top.2007.03.007","DOIUrl":"10.1016/j.top.2007.03.007","url":null,"abstract":"<div><p>We study the behavior of the Nil-subgroups of <span><math><mi>K</mi></math></span>-groups under localization. As a consequence of our results, we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character, we obtain a complete computation of the rationalized source of the <span><math><mi>K</mi></math></span>-theoretic assembly map that appears in the Farrell–Jones conjecture in terms of group homology and the <span><math><mi>K</mi></math></span>-groups of finite cyclic subgroups.</p><p>Specifically we prove that under mild assumptions we can always write the Nil-groups and End-groups of the localized ring as a certain colimit over the Nil-groups and End-groups of the ring, generalizing a result of Vorst. We define Frobenius and Verschiebung operations on certain Nil-groups. These operations provide the tool to prove that Nil-groups are modules over the ring of Witt-vectors and are either trivial or not finitely generated as Abelian groups. Combining the localization results with the Witt-vector module structure, we obtain that Nil and localization at an appropriate multiplicatively closed set <span><math><mi>S</mi></math></span> commute, i.e. <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mstyle><mi>Nil</mi></mstyle><mo>=</mo><mstyle><mi>Nil</mi></mstyle><mspace></mspace><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. An important corollary is that the Nil-groups appearing in the decomposition of the <span><math><mi>K</mi></math></span>-groups of virtually cyclic groups are torsion groups.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 3","pages":"Pages 160-202"},"PeriodicalIF":0.0,"publicationDate":"2008-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.03.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188652","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":"Subgroup separability and virtual retractions of groups","authors":"D.D. Long ,&nbsp;A.W. Reid","doi":"10.1016/j.top.2006.04.001","DOIUrl":"10.1016/j.top.2006.04.001","url":null,"abstract":"<div><p>We discuss separability properties of discrete groups, and introduce a new property of groups that is motivated by a geometric proof of separability of geometrically finite subgroups of Kleinian groups. This property appears natural in that it provides a general framework for old questions in the geometry and topology of hyperbolic manifolds and discrete groups.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 3","pages":"Pages 137-159"},"PeriodicalIF":0.0,"publicationDate":"2008-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.04.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188487","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":"Publication information","authors":"","doi":"10.1016/S0040-9383(08)00004-9","DOIUrl":"https://doi.org/10.1016/S0040-9383(08)00004-9","url":null,"abstract":"","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 3","pages":"Page IFC"},"PeriodicalIF":0.0,"publicationDate":"2008-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0040-9383(08)00004-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137341312","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":"3-manifolds which are orbit spaces of diffeomorphisms","authors":"C. Bonatti,&nbsp;L. Paoluzzi","doi":"10.1016/j.top.2007.06.003","DOIUrl":"10.1016/j.top.2007.06.003","url":null,"abstract":"<div><p>In a very general setting, we show that a 3-manifold obtained as the orbit space of the basin of a topological attractor is either <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> or irreducible.</p><p>We then study in more detail the topology of a class of 3-manifolds which are also orbit spaces and arise as invariants of gradient-like diffeomorphisms (in dimension 3). Up to a finite number of exceptions, which we explicitly describe, all these manifolds are Haken and, by changing the diffeomorphism by a finite power, all the Seifert components of the Jaco–Shalen–Johannson decomposition of these manifolds are made into product circle bundles.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 2","pages":"Pages 71-100"},"PeriodicalIF":0.0,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.06.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188724","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":"Publication information","authors":"","doi":"10.1016/S0040-9383(07)00085-7","DOIUrl":"https://doi.org/10.1016/S0040-9383(07)00085-7","url":null,"abstract":"","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 2","pages":"Page IFC"},"PeriodicalIF":0.0,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0040-9383(07)00085-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"92142914","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":"Complex hyperbolic Fenchel–Nielsen coordinates","authors":"John R. Parker,&nbsp;Ioannis D. Platis","doi":"10.1016/j.top.2007.08.001","DOIUrl":"10.1016/j.top.2007.08.001","url":null,"abstract":"<div><p>Let <span><math><mi>Σ</mi></math></span> be a closed, orientable surface of genus <span><math><mi>g</mi></math></span>. It is known that the <span><math><mstyle><mi>SU</mi></mstyle><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span> representation variety of <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>Σ</mi><mo>)</mo></mrow></math></span> has <span><math><mn>2</mn><mi>g</mi><mo>−</mo><mn>3</mn></math></span> components of (real) dimension <span><math><mn>16</mn><mi>g</mi><mo>−</mo><mn>16</mn></math></span> and two components of dimension <span><math><mn>8</mn><mi>g</mi><mo>−</mo><mn>6</mn></math></span>. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a subset of the representation variety that contains the quasi-Fuchsian representations. These coordinates are a natural generalisation of Fenchel–Nielsen coordinates on the Teichmüller space of <span><math><mi>Σ</mi></math></span> and complex Fenchel–Nielsen coordinates on the (classical) quasi-Fuchsian space of <span><math><mi>Σ</mi></math></span>.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 2","pages":"Pages 101-135"},"PeriodicalIF":0.0,"publicationDate":"2008-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.08.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188758","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":"Integrals of equivariant forms and a Gauss–Bonnet theorem for constructible sheaves","authors":"Matvei Libine","doi":"10.1016/j.top.2006.07.001","DOIUrl":"10.1016/j.top.2006.07.001","url":null,"abstract":"<div><p>The Berline–Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539–541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>.</p><p>As an application of this generalization, we prove an analogue of the Gauss–Bonnet theorem for constructible sheaves. If <span><math><mi>F</mi></math></span> is a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>-equivariant sheaf on a complex projective manifold <span><math><mi>M</mi></math></span>, then the Euler characteristic of <span><math><mi>M</mi></math></span> with respect to <span><math><mi>F</mi></math></span><span><span><span><math><mi>χ</mi><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mo>)</mo></mrow></mrow><mrow><munder><mrow><mo>dim</mo></mrow><mrow><mi>C</mi></mrow></munder><mi>M</mi></mrow></msup></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><mstyle><mi>Ch</mi></mstyle><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></msub><mover><mrow><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow></msub></mrow><mrow><mo>˜</mo></mrow></mover></math></span></span></span> as distributions on <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>, where <span><math><mstyle><mi>Ch</mi></mstyle><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></math></span> is the characteristic cycle of <span><math><mi>F</mi></math></span> and <span><math><mover><mrow><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>C</mi></mrow></msub></mrow></msub></mrow><mrow><mo>˜</mo></mrow></mover></math></span> is the Euler form of <span><math><mi>M</mi></math></span> extended to the cotangent space <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>∗</mo></mrow></msup><mi>M</mi></math></span> (independently of <span><math><mi>F</mi></math></span>). We also consider an analogue of Duistermaat–Heckman measures for real reductive Lie groups acting on symplectic manifolds.</p><p>In [M. Libine, Riemann–Roch–Hirzebruch integral formula for characters of reductive Lie groups, Represent. Theory 9 (2005) 507–524. Also <span>math.RT/0312454</span><svg><path></path></svg>] I apply the results of this article to obtain a Riemann–Roch–Hirzebruch type integral formula for characters of representations of reductive groups.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 1","pages":"Pages 1-39"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2006.07.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188090","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 trace formula for the forcing relation of braids","authors":"Boju Jiang,&nbsp;Hao Zheng","doi":"10.1016/j.top.2007.06.002","DOIUrl":"10.1016/j.top.2007.06.002","url":null,"abstract":"<div><p>The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 1","pages":"Pages 51-70"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.06.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188714","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":"Publication information","authors":"","doi":"10.1016/S0040-9383(07)00077-8","DOIUrl":"https://doi.org/10.1016/S0040-9383(07)00077-8","url":null,"abstract":"","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 1","pages":"Page IFC"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0040-9383(07)00077-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"137374027","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":"Surfaces in 4-manifolds and their mapping class groups","authors":"Susumu Hirose ,&nbsp;Akira Yasuhara","doi":"10.1016/j.top.2007.05.001","DOIUrl":"10.1016/j.top.2007.05.001","url":null,"abstract":"<div><p>A surface in a smooth 4-manifold is called <em>flexible</em> if, for any diffeomorphism <span><math><mi>ϕ</mi></math></span> on the surface, there is a diffeomorphism on the 4-manifold whose restriction on the surface is <span><math><mi>ϕ</mi></math></span> and which is isotopic to the identity. We investigate a sufficient condition for a smooth 4-manifold <span><math><mi>M</mi></math></span> to include flexible knotted surfaces, and introduce a local operation in simply connected 4-manifolds for obtaining a flexible knotted surface from any knotted surface.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 1","pages":"Pages 41-50"},"PeriodicalIF":0.0,"publicationDate":"2008-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.05.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188663","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}