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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ 3$-pseudomanifolds with preassigned links

Author: Amos Altshuler
Journal: Trans. Amer. Math. Soc. 241 (1978), 213-237
MSC: Primary 57N10; Secondary 57Q05
MathSciNet review: 492298
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Abstract: A 3-pseudomanifold is a finite connected simplicial 3-complex $ \mathcal{K}$ such that every triangle in $ \mathcal{K}$ belongs to precisely two 3-simplices of $ \mathcal{K}$, the link of every edge in $ \mathcal{K}$ is a circuit, and the link of every vertex in $ \mathcal{K}$ is a closed 2-manifold. It is proved that for every finite set $ \sum $ of closed 2-manifolds, there exists a 3-pseudomanifold $ \mathcal{K}$ such that the link of every vertex in $ \mathcal{K}$ is homeomorphic to some $ S\, \in \,\sum $, and every $ S\, \in \,\sum $ is homeomorphic to the link of some vertex in $ \mathcal{K}$.

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  • [1] James W. Alexander, The combinatorial theory of complexes, Ann. of Math. (2) 31 (1930), no. 2, 292–320. MR 1502943, 10.2307/1968099
  • [2] P. S. Aleksandrov, Combinatorial topology. Vol. 1, Graylock Press, Rochester, N. Y., 1956. MR 0076324
  • [3] Amos Altshuler, Combinatorial 3-manifolds with few vertices, J. Combinatorial Theory Ser. A 16 (1974), 165–173. MR 0346797
  • [4] David Barnette, Graph theorems for manifolds, Israel J. Math. 16 (1973), 62–72. MR 0360364
  • [5] P. Franklin, A six color problem, J. Mathematical Phys. 13 (1934), 363-369.
  • [6] Branko Grünbaum, Convex polytopes, With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. MR 0226496
  • [7] -, Regularity of graphs, complexes and designs, Comptes Rendus Colloq. Internat. C.N.R.S., Problèmes Combinatoires et Théorie des Graphes, Paris, July 1976 (to appear).
  • [8] P. J. Hilton and S. Wylie, Homology theory, Cambridge Univ. Press, Cambridge, 1967.
  • [9] P. McMullen, The number of neighbourly 𝑑-polytopes with 𝑑+3 vertices, Mathematika 21 (1974), 26–31. MR 0367812
  • [10] Udo Pachner, Untersuchungen über die Dualität zwischen Schnitten und Submannigfaltigkeiten konvexer Polytope und allgemeinerer kombinatorischer Mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamburg 50 (1980), 40–56 (German). MR 593735, 10.1007/BF02941413
  • [11] -, Schnitt- und Überdeckungszahlen kombinatorischer Sphären, Geometriae Dedicata (to appear).

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Keywords: 3-pseudomanifold, simplicial complex, 2-manifold, 3-manifold, link, star, assembling, neighborly complex, f-vector, convex polytope, automorphism-group, finite field, projective transformation, nerve-graph, Gale-diagram
Article copyright: © Copyright 1978 American Mathematical Society