Characteristic Class of Isotopy for Surfaces

Yuxue Ren, Chengfeng Wen, Shengxian Zhen, Na Lei, Feng Luo, David Xianfeng Gu

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2139-2156
Number of pages18
JournalJournal of Systems Science and Complexity
Issue number6
StatePublished - Dec 2020

All Science Journal Classification (ASJC) codes

  • Computer Science (miscellaneous)
  • Information Systems


  • Characteristic class
  • Haefliger-Wu invariants
  • cohomology
  • embedding
  • isotopy


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