A peculiar triangulation of the 3-sphere

Altshuler, A.

doi:10.1090/S0002-9939-1976-0397744-3

A peculiar triangulation of the $3$-sphere
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Proc. Amer. Math. Soc. 54 (1976), 449-452 Request permission

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

James W. Alexander, The combinatorial theory of complexes, Ann. of Math. (2) 31 (1930), no. 2, 292–320. MR 1502943, DOI 10.2307/1968099
Amos Altshuler, Combinatorial $3$-manifolds with few vertices, J. Combinatorial Theory Ser. A 16 (1974), 165–173. MR 346797, DOI 10.1016/0097-3165(74)90042-9
Amos Altshuler and Leon Steinberg, Neighborly combinatorial $3$-manifolds with $9$ vertices, Discrete Math. 8 (1974), 113–137. MR 341491, DOI 10.1016/0012-365X(74)90059-4

An enumeration of combinatorial

manifolds with

vertices

L. W. Beineke and R. E. Pippert, The number of labeled dissections of a $k$-ball, Math. Ann. 191 (1971), 87–98. MR 282860, DOI 10.1007/BF02330563

Geometrical combinatorial topology

On the enumeration of convex polytopes and combinatorial spheres

P. J. Hilton and S. Wylie, Homology theory: An introduction to algebraic topology, Cambridge University Press, New York, 1960. MR 0115161
P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, London Mathematical Society Lecture Note Series, vol. 3, Cambridge University Press, London-New York, 1971. Prepared in collaboration with J. E. Reeve and A. A. Ball. MR 0301635
Gopal Danaraj and Victor Klee, Shellings of spheres and polytopes, Duke Math. J. 41 (1974), 443–451. MR 345113

Additional Information

Journal: Proc. Amer. Math. Soc. 54 (1976), 449-452
DOI: https://doi.org/10.1090/S0002-9939-1976-0397744-3
MathSciNet review: 0397744