Legendrian non-isotopic unit conormal bundles in high dimensions

IF 1.1 2区 数学 Q2 MATHEMATICS
Yukihiro Okamoto
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引用次数: 0

Abstract

For any compact connected submanifold K $K$ of R n $\mathbb {R}^n$ , let Λ K $\Lambda _K$ denote its unit conormal bundle, which is a Legendrian submanifold of the unit cotangent bundle of R n $\mathbb {R}^n$ . In this paper, we give examples of pairs ( K 0 , K 1 ) $(K_0,K_1)$ of compact connected submanifolds of R n $\mathbb {R}^n$ such that Λ K 0 $\Lambda _{K_0}$ is not Legendrian isotopic to Λ K 1 $\Lambda _{K_1}$ , although they cannot be distinguished by classical invariants. Here, K 1 $K_1$ is the image of an embedding ι f : K 0 R n $\iota _f \colon K_0 \rightarrow \mathbb {R}^n$ which is regular homotopic to the inclusion map of K 0 $K_0$ and the codimension in R n $\mathbb {R}^n$ 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 Λ K $\Lambda _K$ when the codimension of K $K$ is greater than or equal to 4. The main examples Λ K 0 $\Lambda _{K_0}$ and Λ K 1 $\Lambda _{K_1}$ are distinguished by the coproduct, which is computed by using an idea of string topology.

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高维的勒让德非同位素单位正法线束
对于任意紧连通子流形K $K$ R的n次方 $\mathbb {R}^n$ ,让Λ K $\Lambda _K$ 表示它的单位法向束,它是rn的单位余切束的legendian子流形 $\mathbb {R}^n$ 。本文给出了(k0, k1)对的例子。 $(K_0,K_1)$ rn的紧连通子流形 $\mathbb {R}^n$ 使得Λ k0 $\Lambda _{K_0}$ 难道不是Legendrian的同位素Λ k1吗 $\Lambda _{K_1}$ ,尽管它们不能用经典不变量来区分。这里是k1 $K_1$ 是嵌入函数的图像f: K 0→R n $\iota _f \colon K_0 \rightarrow \mathbb {R}^n$ 哪个是k0的包含映射的正则同伦 $K_0$ 和rn中的余维 $\mathbb {R}^n$ 大于等于4。作为非经典不变量,在一定条件下,我们定义了条形Legendrian接触同调及其上的一个余积。然后,我们给出Λ K的这些不变量的纯拓扑描述 $\Lambda _K$ 当K的余维 $K$ 大于等于4。主要的例子Λ K 0 $\Lambda _{K_0}$ 和Λ k1 $\Lambda _{K_1}$ 是由余积来区分的,余积是用弦拓扑的思想计算出来的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Topology
Journal of Topology 数学-数学
CiteScore
2.00
自引率
9.10%
发文量
62
审稿时长
>12 weeks
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.
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