TopologyPub Date : 2008-11-01DOI: 10.1016/j.top.2008.04.001
Henryk Żołądek
{"title":"An application of Newton–Puiseux charts to the Jacobian problem","authors":"Henryk Żołądek","doi":"10.1016/j.top.2008.04.001","DOIUrl":"10.1016/j.top.2008.04.001","url":null,"abstract":"<div><p>We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new analytical and topological properties. We prove that the Jacobian Conjecture holds true for maps <span><math><mrow><mo>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></math></span> whose topological degree is <span><math><mo>≤</mo><mn>5</mn></math></span>, for maps with <span><math><mo>gcd</mo><mrow><mo>(</mo><mo>deg</mo><mi>f</mi><mo>,</mo><mo>deg</mo><mi>g</mi><mo>)</mo></mrow><mo>≤</mo><mn>16</mn></math></span> and for maps with. <span><math><mo>gcd</mo><mrow><mo>(</mo><mo>deg</mo><mi>f</mi><mo>,</mo><mo>deg</mo><mi>g</mi><mo>)</mo></mrow></math></span> equal to 2 times a prime.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 6","pages":"Pages 431-469"},"PeriodicalIF":0.0,"publicationDate":"2008-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2008.04.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188825","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 : 2008-11-01DOI: 10.1016/j.top.2007.09.002
Stephen D. Theriault
{"title":"Homotopy exponents of mod2r Moore spaces","authors":"Stephen D. Theriault","doi":"10.1016/j.top.2007.09.002","DOIUrl":"10.1016/j.top.2007.09.002","url":null,"abstract":"<div><p>We prove that <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>⋅</mo><msub><mrow><mi>π</mi></mrow><mrow><mo>∗</mo></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></span> provided <span><math><mi>m</mi><mo>≥</mo><mn>4</mn></math></span> and <span><math><mi>r</mi><mo>≥</mo><mn>6</mn></math></span>. This is the best possible result. As well, for <span><math><mn>2</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mn>5</mn></math></span> we obtain upper bounds on the homotopy exponent of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 6","pages":"Pages 369-398"},"PeriodicalIF":0.0,"publicationDate":"2008-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.09.002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188790","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 : 2008-09-01DOI: 10.1016/j.top.2007.09.003
Robert Young
{"title":"Averaged Dehn functions for nilpotent groups","authors":"Robert Young","doi":"10.1016/j.top.2007.09.003","DOIUrl":"10.1016/j.top.2007.09.003","url":null,"abstract":"<div><p>Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality <span><math><mi>δ</mi><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo><</mo><mi>C</mi><msup><mrow><mi>l</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span> for <span><math><mi>α</mi><mo>></mo><mn>2</mn></math></span>, then it satisfies the averaged isoperimetric inequality <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mstyle><mi>avg</mi></mstyle></mrow></msup><mrow><mo>(</mo><mi>l</mi><mo>)</mo></mrow><mo><</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span>. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 5","pages":"Pages 351-367"},"PeriodicalIF":0.0,"publicationDate":"2008-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.09.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188814","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 : 2008-09-01DOI: 10.1016/j.top.2007.03.006
Sungmo Kang
{"title":"Reducible and toroidal Dehn fillings with distance 3","authors":"Sungmo Kang","doi":"10.1016/j.top.2007.03.006","DOIUrl":"10.1016/j.top.2007.03.006","url":null,"abstract":"<div><p>If a simple 3-manifold <span><math><mi>M</mi></math></span> admits a reducible and a toroidal Dehn filling, the distance between the filling slopes is known to be bounded by three. In this paper, we classify all manifolds which admit a reducible Dehn filling and a toroidal Dehn filling with distance 3.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 5","pages":"Pages 277-315"},"PeriodicalIF":0.0,"publicationDate":"2008-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.03.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188641","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 : 2008-09-01DOI: 10.1016/j.top.2007.06.001
Steven B. Bradlow , Oscar García-Prada , Peter B. Gothen
{"title":"Homotopy groups of moduli spaces of representations","authors":"Steven B. Bradlow , Oscar García-Prada , Peter B. Gothen","doi":"10.1016/j.top.2007.06.001","DOIUrl":"10.1016/j.top.2007.06.001","url":null,"abstract":"<div><p>We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups <span><math><mstyle><mi>GL</mi></mstyle><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>C</mi><mo>)</mo></mrow></math></span> and <span><math><mstyle><mi>U</mi></mstyle><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>. Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott–Morse theory on the corresponding moduli spaces.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 4","pages":"Pages 203-224"},"PeriodicalIF":0.0,"publicationDate":"2008-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.06.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188689","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 : 2008-09-01DOI: 10.1016/j.top.2007.07.001
Hiroshi Iritani
{"title":"Quantum D-modules and generalized mirror transformations","authors":"Hiroshi Iritani","doi":"10.1016/j.top.2007.07.001","DOIUrl":"10.1016/j.top.2007.07.001","url":null,"abstract":"<div><p>In the previous paper [Hiroshi Iritani, Quantum <span><math><mi>D</mi></math></span>-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum <span><math><mi>D</mi></math></span>-module after a mirror transformation when the first Chern class <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></math></span> of the tangent bundle is nef. In this paper, even when <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></math></span> is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum <span><math><mi>D</mi></math></span>-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 4","pages":"Pages 225-276"},"PeriodicalIF":0.0,"publicationDate":"2008-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.07.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188741","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 : 2008-09-01DOI: 10.1016/j.top.2007.09.001
Olivier Couture
{"title":"Strongly invertible links and divides","authors":"Olivier Couture","doi":"10.1016/j.top.2007.09.001","DOIUrl":"10.1016/j.top.2007.09.001","url":null,"abstract":"<div><p>To a proper generic immersion of a finite number of copies of the unit interval in a 2-disc, called a divide, A’Campo associates a link in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. From the more general notion of ordered Morse signed divides, one obtains a braid presentation of links of divides. In this paper, we prove that every strongly invertible link is isotopic to the link of an ordered Morse signed divide. We give fundamental moves for ordered Morse signed divides and show that strongly invertible links are equivalent if and only if we can pass from one ordered Morse signed divide to the other by a sequence of such moves. Then we associate a polynomial to an ordered Morse signed divide, invariant for these moves. So this polynomial is invariant for the equivalence of strongly invertible links.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"47 5","pages":"Pages 316-350"},"PeriodicalIF":0.0,"publicationDate":"2008-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2007.09.001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"55188778","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}
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