Abstract
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.
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References
Wu W, Science record, N.S., 1959, 3: 342–351.
Wu W, A Theory of Embedding, Immersion, and Isotopy of Polytopes in an Euclidean Space, Science Press, Beijing, 1965.
Wu W, On the realization of complexes in Euclidean spaces I, Selected Works of Wen-Tsun Wu, World Scientific, 2008, 23–69.
Haeiger A, Differentiable links, Topology, 1962, 1(3): 241–244.
Haeiger A, Knotted (4k — 1)-spheres in 6k-space, Annals of Mathematics, 1962, 75: 452–466.
Haeiger A, Plongements differentiables de varietes dans varietes, Commentarii Mathematici Helvetici, 1962, 36(1): 47–82.
Haeiger A, Differentiable embeddings of Sn in Sn−q for q> 2, Annals of Mathematics, 1966, 83: 402–436.
Crowley D and Skopenkov A, A classification of embeddings of non-simply connected 4-manifolds in 7-space, Alexandroff Readings, 2016.
Kreck M, Isotopy classes of diffeomorphisms of (k — 1)-connected almost-parallelizable 2k-manifolds, Algebraic Topology Aarhus, 1978, 643–663.
Skopenkov A, A classification of smooth embeddings of 4-manifolds in 7-space, I, Topology and its Applications, 2010, 157: 2094–2110.
Skopenkov A, A characterization of submanifolds by a homogeneity condition, Topology and its Applications, 2007, 154: 1894–1897.
Skopenkov A, On the Haeiger-Hirsch-Wu invariants for embeddings and immersions, Commentarii Mathematici Helvetici, 2002, 77(1): 78–124.
Skopenkov A, A classification of smooth embeddings of 3-manifolds in 6-space, Mathematische Zeitschrift, 2008, 260: 647–672.
Skopenkov A, Embeddings of k-Connected n-Manifolds into ⋄2n−k−1, Proceedings of the American Mathematical Society, 2010, 138(9): 3377–3389.
Gonçcalves D and Skopenkov A, Embeddings of homology equivalent manifolds with boundary, Topology and Its Applications, 2006, 153(12): 2026–2034.
Tonkonog D, Embedding 3-manifolds with boundary into closed 3-manifolds, Topology and Its Applications, 2011, 158(9): 1157–1162.
Skopenkov A, Embedding and knotting of manifolds in euclidean spaces, London Mathematical Society Lecture Notes, 2008, 347: 248–342.
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This research was supported by the Science Challenge Project of China (TZZT2019-B1) and the National Natural Science Foundation of China under Grant Nos. 61720106005, 61772105, 61936002.
This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
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Ren, Y., Wen, C., Zhen, S. et al. Characteristic Class of Isotopy for Surfaces. J Syst Sci Complex 33, 2139–2156 (2020). https://doi.org/10.1007/s11424-020-9053-8
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DOI: https://doi.org/10.1007/s11424-020-9053-8