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Isomorphism between Tangle and Cobordism Double Categories

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1765)

Abstract

In the previous two chapters we introduced two double categories. One is the double category of framed relative cobordisms \( \widetilde{Cob} \) from Chapter 1, and the other is the double category of equivalence classes of admissible tangles Tgl from Chapter 2. The purpose of this chapter is to show that they are isomorphic as double categories. This provides us with a systematic combinatorial presentation of 3-manifolds with comers that respects the two gluing operations. The main result of the first three chapters of this book is summarized in the following presentation theorem.

Keywords

  • Morse Function
  • Critical Manifold
  • Link Diagram
  • Cobordism Class
  • Standard Diagram

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Isomorphism between Tangle and Cobordism Double Categories. In: Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners. Lecture Notes in Mathematics, vol 1765. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44625-7_4

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  • DOI: https://doi.org/10.1007/3-540-44625-7_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42416-1

  • Online ISBN: 978-3-540-44625-5

  • eBook Packages: Springer Book Archive