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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Klein bottles in circle bundles


Author: John W. Wood
Journal: Proc. Amer. Math. Soc. 28 (1971), 607-608
MSC: Primary 57.20
DOI: https://doi.org/10.1090/S0002-9939-1971-0278324-2
MathSciNet review: 0278324
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Abstract: We prove that the Klein bottle embeds in the total space $ E$ of an orientable $ {S^1}$-bundle over an orientable $ 2$-manifold $ M$ if and only if $ M = {S^2}$ and $ E = {S^1} \times {S^2}$ or the lens space $ L(4,1)$.


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

  • [1] G. E. Bredon and J. W. Wood, Non-orientable surfaces in orientable $ 3$-manifolds, Invent. Math. 7 (1969), 83-110. MR 39 #7616. MR 0246312 (39:7616)
  • [2] P. J. Hilton and S. Wylie, Homology theory: An introduction to algebraic topology, Cambridge Univ. Press, New York, 1960. MR 22 #5963. MR 0115161 (22:5963)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0278324-2
Keywords: Embedding, Klein bottle, $ {S^1}$-bundle over $ 2$-manifold, lens space
Article copyright: © Copyright 1971 American Mathematical Society

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