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

 
 

 

A peculiar triangulation of the $ 3$-sphere


Author: A. Altshuler
Journal: Proc. Amer. Math. Soc. 54 (1976), 449-452
DOI: https://doi.org/10.1090/S0002-9939-1976-0397744-3
MathSciNet review: 0397744
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Abstract | References | Additional Information

Abstract: A triangulation of the $ 3$-sphere with $ 10$ vertices is presented, which is not directly obtainable by a generalized stellar subdivision from any $ 3$-sphere with $ 9$ vertices. This answers in the affirmative a conjecture by B. Grünbaum. This example is shown to be the minimal possible.


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

  • [1] J. W. Alexander, The combinatorial theory of complexes, Ann. of Math. 31 (1930), 292-320. MR 1502943
  • [2] A. Altshuler, Combinatorial $ 3$-manifolds with few vertices, J. Combinatorial Theory (A) 16 (1974), 165-173. MR 0346797 (49:11521)
  • [3] A. Altshuler and L. Steinberg, Neighborly combinatorial $ 3$-manifolds with $ 9$ vertices, Discrete Math. 8 (1974), 113-137. MR 0341491 (49:6242)
  • [4] -, An enumeration of combinatorial $ 3$-manifolds with $ 9$ vertices, Discrete Math. (to appear).
  • [5] L. W. Beineke and R. E. Pippert, The number of labelled dissections of a $ k$-ball, Math. Ann. 191 (1971), 87-98. MR 44 #94. MR 0282860 (44:94)
  • [6] L. C. Glaser, Geometrical combinatorial topology. I, Van Nostrand Reinhold, New York, 1970.
  • [7] B. Grünbaum, On the enumeration of convex polytopes and combinatorial spheres, ONR Technical Report, University of Washington, May 1969.
  • [8] 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)
  • [9] P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, London Math. Soc. Lecture Note Series, 3, Cambridge Univ. Press, London and New York, 1971. MR 46 #791. MR 0301635 (46:791)
  • [10] G. Danaraj and V. Klee, Shellings of spheres and polytopes, Duke Math. J. 41 (1974), 443-451. MR 0345113 (49:9852)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0397744-3
Keywords: (Abstract) simplicial $ n$-complex, combinatorial $ n$-sphere, directly obtainable, shellable sphere
Article copyright: © Copyright 1976 American Mathematical Society

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