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An extension of Rourke's proof that $ \Omega\sb 3=0$ to nonorientable manifolds


Authors: Fredric D. Ancel and Craig R. Guilbault
Journal: Proc. Amer. Math. Soc. 115 (1992), 283-291
MSC: Primary 57N10; Secondary 57N70
DOI: https://doi.org/10.1090/S0002-9939-1992-1092913-0
MathSciNet review: 1092913
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Abstract: A classical result in manifold theory states that every closed $ 3$-manifold bounds a compact $ 4$-manifold. In 1985 C. Rourke discovered a strikingly short and elementary proof of the orientable case of this theorem $ ({\Omega _3} = 0)$. In this note we show that Rourke's approach can be extended to include nonorientable $ 3$-manifolds.


References [Enhancements On Off] (What's this?)

  • [1] W. B. R. Lickorish, A representation of orientable combinatorial three-manifolds, Ann. of Math. (2) 76 (1962), 531-540. MR 0151948 (27:1929)
  • [2] -, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307-317. MR 0145498 (26:3029)
  • [3] V. A. Rokhlin, A $ 3$-dimensional manifold is the boundary of a $ 4$-dimensional manifold, Dokl. Akad. Nauk. SSSR 81 (1951), 355.
  • [4] C. Rourke, A new proof that $ {\Omega _3} = 0$, J. London Math. Soc. (2) 31 (1985), 373-376. MR 809959 (87f:57016)
  • [5] R. Thom, Quelques propriétés globales des variétés différentiables, Comm. Math. Helv. 28 (1954), 17-86. MR 0061823 (15:890a)
  • [6] A. H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960), 503-528. MR 0125588 (23:A2887)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1092913-0
Keywords: Surface, $ 3$-manifold, $ 4$-manifold, nonorientable, bordism, cobordism, Dehn surgery, Heegaard diagram
Article copyright: © Copyright 1992 American Mathematical Society

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