{"title":"Legendrian non-isotopic unit conormal bundles in high dimensions","authors":"Yukihiro Okamoto","doi":"10.1112/topo.70039","DOIUrl":null,"url":null,"abstract":"<p>For any compact connected submanifold <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math>, let <span></span><math>\n <semantics>\n <msub>\n <mi>Λ</mi>\n <mi>K</mi>\n </msub>\n <annotation>$\\Lambda _K$</annotation>\n </semantics></math> denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math>. In this paper, we give examples of pairs <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>K</mi>\n <mn>1</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(K_0,K_1)$</annotation>\n </semantics></math> of compact connected submanifolds of <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <msub>\n <mi>Λ</mi>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n </msub>\n <annotation>$\\Lambda _{K_0}$</annotation>\n </semantics></math> is not Legendrian isotopic to <span></span><math>\n <semantics>\n <msub>\n <mi>Λ</mi>\n <msub>\n <mi>K</mi>\n <mn>1</mn>\n </msub>\n </msub>\n <annotation>$\\Lambda _{K_1}$</annotation>\n </semantics></math>, although they cannot be distinguished by classical invariants. Here, <span></span><math>\n <semantics>\n <msub>\n <mi>K</mi>\n <mn>1</mn>\n </msub>\n <annotation>$K_1$</annotation>\n </semantics></math> is the image of an embedding <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>ι</mi>\n <mi>f</mi>\n </msub>\n <mo>:</mo>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n <mo>→</mo>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n </mrow>\n <annotation>$\\iota _f \\colon K_0 \\rightarrow \\mathbb {R}^n$</annotation>\n </semantics></math> which is regular homotopic to the inclusion map of <span></span><math>\n <semantics>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n <annotation>$K_0$</annotation>\n </semantics></math> and the codimension in <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>$\\mathbb {R}^n$</annotation>\n </semantics></math> is greater than or equal to 4. As non-classical invariants, we define the strip Legendrian contact homology and a coproduct on it under certain conditions on Legendrian submanifolds. Then, we give a purely topological description of these invariants for <span></span><math>\n <semantics>\n <msub>\n <mi>Λ</mi>\n <mi>K</mi>\n </msub>\n <annotation>$\\Lambda _K$</annotation>\n </semantics></math> when the codimension of <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> is greater than or equal to 4. The main examples <span></span><math>\n <semantics>\n <msub>\n <mi>Λ</mi>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n </msub>\n <annotation>$\\Lambda _{K_0}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <msub>\n <mi>Λ</mi>\n <msub>\n <mi>K</mi>\n <mn>1</mn>\n </msub>\n </msub>\n <annotation>$\\Lambda _{K_1}$</annotation>\n </semantics></math> are distinguished by the coproduct, which is computed by using an idea of string topology.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"18 3","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/topo.70039","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Topology","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/topo.70039","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For any compact connected submanifold of , let denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of . In this paper, we give examples of pairs of compact connected submanifolds of such that is not Legendrian isotopic to , although they cannot be distinguished by classical invariants. Here, is the image of an embedding which is regular homotopic to the inclusion map of and the codimension in is greater than or equal to 4. As non-classical invariants, we define the strip Legendrian contact homology and a coproduct on it under certain conditions on Legendrian submanifolds. Then, we give a purely topological description of these invariants for when the codimension of is greater than or equal to 4. The main examples and are distinguished by the coproduct, which is computed by using an idea of string topology.
期刊介绍:
The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal.
The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.