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Proceedings of the American Mathematical Society

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

 
 

 

A characterisation of punctured open $ 3$-cells


Authors: O. L. Costich, P. H. Doyle and D. E. Galewski
Journal: Proc. Amer. Math. Soc. 28 (1971), 295-298
MSC: Primary 54.78
DOI: https://doi.org/10.1090/S0002-9939-1971-0271919-1
MathSciNet review: 0271919
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Abstract | References | Similar Articles | Additional Information

Abstract: A proof is given using standard methods of the topology of three-dimensional manifolds of the following characterization of punctured cubes: A connected, open $ 3$-manifold $ M$ is topological $ {E^3}$ with $ k$ points removed if and only if every polyhedral simple closed curve in $ M$ lies in a topological cube in $ M$ and the rank of $ {\pi _2}(M)$ is $ k$. An application is given.


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

  • [1] R. H. Bing, Necessary and sufficient conditions that a $ 3$-manifold be $ {S^3}$, Ann. of Math. (2) 68 (1958), 17-37. MR 20 #1973. MR 0095471 (20:1973)
  • [2] M. Brown, The monotone union of open $ n$-cells is an open $ n$-cell, Proc. Amer. Math. Soc. 12 (1961), 812-814. MR 23 #A4129. MR 0126835 (23:A4129)
  • [3] D. R. McMillan, Jr., Cartesian products of contractible open manifolds, Bull. Amer. Math. Soc. 67 (1961), 510-514. MR 24 #A1132. MR 0131280 (24:A1132)
  • [4] J. H. C. Whitehead, On $ 2$-spheres in $ 3$-manifolds, Bull. Amer. Math. Soc. 64 (1958), 161-166. MR 21 #2241. MR 0103473 (21:2241)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0271919-1
Keywords: Three-dimensional manifold, punctured cube, irreducible manifold
Article copyright: © Copyright 1971 American Mathematical Society

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