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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
DOI: https://doi.org/10.1090/S0002-9947-1978-0492298-1
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}$.


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

  • [1] J. W. Alexander, The combinatorial theory of complexes, Ann. of Math. (2) 31 (1930), 292-320. MR 1502943
  • [2] P. S. Aleksandrov, Combinatorial topology, Vol. 1, Graylock Press, Rochester, N. Y., 1956. MR 0076324 (17:882a)
  • [3] A. Altshuler, Combinatorial 3-manifolds with few vertices, J. Combinatorial Theory Ser. A 16 (1974), 165-173. MR 0346797 (49:11521)
  • [4] D. Barnette, Graph theorems for manifolds, Israel J. Math. 16 (1973),62-72. MR 0360364 (50:12814)
  • [5] P. Franklin, A six color problem, J. Mathematical Phys. 13 (1934), 363-369.
  • [6] B. Grünbaum, Convex polytopes, Wiley, New York, 1967. MR 0226496 (37:2085)
  • [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 (51:4054)
  • [10] U. Pachner, Untersuchungen über die Dualität zwischen Schnitten und Submannigfaltigkeiten knovexer polytope und Allgemeinerer kombinatorischer Mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamburg (to appear). MR 593735 (82e:57007)
  • [11] -, Schnitt- und Überdeckungszahlen kombinatorischer Sphären, Geometriae Dedicata (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0492298-1
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

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