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Characteristic Class of Isotopy for Surfaces
Journal of Systems Science and Complexity ( IF 2.1 ) Pub Date : 2020-08-04 , DOI: 10.1007/s11424-020-9053-8
Yuxue Ren , Chengfeng Wen , Shengxian Zhen , Na Lei , Feng Luo , David Xianfeng Gu

It is an important problem in topology to verify whether two embeddings are isotopic. This work proposes an algorithm for computing Haefliger-Wu invariants for isotopy based on algebraic topological methods. Given a simplicial complex embedded in the Euclidean space, the deleted product of it is the direct product with diagonal removed. The Gauss map transforms the deleted product to the unit sphere. The pull-back of the generator of the cohomology group of the sphere defines characteristic class of the isotopy of the embedding. By using Mayer Vietoris sequence and Künneth theorem, the computational algorithm can be greatly simplified. The authors prove the ranks of homology groups of the deleted product of a closed surface and give explicit construction of the generators of the homology groups of the deleted product. Numerical experimental results show the efficiency and efficacy of the proposed method.



中文翻译:

表面同位素特征分类

验证两个嵌入是否为同位素是拓扑中的一个重要问题。这项工作提出了一种基于代数拓扑方法计算同位素的Haefliger-Wu不变量的算法。给定一个嵌入在欧几里得空间中的简单复形,它的删除乘积就是对角线被删除的直接乘积。高斯贴图将删除的乘积转换为单位球。球的同调群的生成器的后退定义了嵌入同位素的特征类。通过使用Mayer Vietoris序列和Künneth定理,可以大大简化计算算法。作者证明了闭合表面缺失产物的同源基团的等级,并给出了缺失产物同源基团的生成子的显式构造。

更新日期:2020-08-04
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