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The Poincaré conjecture is true in the product of any graph with a disk


Author: David Gillman
Journal: Proc. Amer. Math. Soc. 110 (1990), 829-834
MSC: Primary 57N10; Secondary 57M40, 57Q65
DOI: https://doi.org/10.1090/S0002-9939-1990-1021898-6
MathSciNet review: 1021898
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the only compact $ 3$-manifold-with-boundary which has trivial rational homology, and which embeds in the product of a graph with a disk, is the $ 3$-ball. This implies that no punctured lens space embeds in the product of a graph with a disk. It also implies our title.

The proof relies on a general position argument which enables us to perform surgery.


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

  • [1] D. B. A. Epstein, Embedding punctured manifolds, Proc. Amer. Math. Soc. 16 (1965), 175-176. MR 0208606 (34:8415)
  • [2] D. Gillman and D. Rolfsen, The Zeeman Conjecture for standard spines is equivalent to the Poincaré Conjecture, Topology 22 (1983), 315-323. MR 710105 (85b:57013)
  • [3] E. Moise, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York, 1977. MR 0488059 (58:7631)
  • [4] H. Seifert and W. Threlfall, A textbook of topology, Academic Press, New York, 1980. MR 575168 (82b:55001)
  • [5] E. C. Zeeman, On twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495. MR 0195085 (33:3290)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1021898-6
Article copyright: © Copyright 1990 American Mathematical Society

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