A characterisation of punctured open $3$-cells
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- by O. L. Costich, P. H. Doyle and D. E. Galewski
- Proc. Amer. Math. Soc. 28 (1971), 295-298
- DOI: https://doi.org/10.1090/S0002-9939-1971-0271919-1
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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
- R. H. Bing, Necessary and sufficient conditions that a $3$-manifold be $S^{3}$, Ann. of Math. (2) 68 (1958), 17–37. MR 95471, DOI 10.2307/1970041
- Morton Brown, The monotone union of open $n$-cells is an open $n$-cell, Proc. Amer. Math. Soc. 12 (1961), 812–814. MR 126835, DOI 10.1090/S0002-9939-1961-0126835-6
- D. R. McMillan Jr., Cartesian products of contractible open manifolds, Bull. Amer. Math. Soc. 67 (1961), 510–514. MR 131280, DOI 10.1090/S0002-9904-1961-10662-9
- J. H. C. Whitehead, On $2$-spheres in $3$-manifolds, Bull. Amer. Math. Soc. 64 (1958), 161–166. MR 103473, DOI 10.1090/S0002-9904-1958-10193-7
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- 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