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Manifolds in the Skeletons of Convex Polytopes, Tightness, and Generalized Heawood Inequalities

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Polytopes: Abstract, Convex and Computational

Part of the book series: NATO ASI Series ((ASIC,volume 440))

Abstract

If a 2-dimensional manifold in the 2-dimensional skeleton of a convex d-polytope P contains the 1-skeleton of P then d is bounded in terms of the genus of the surface: this is essentially Heawood’s inequality. In this paper we prove a higher dimensional analogue about 2k-dimensional manifolds containing the k-skeleton of a simplicial convex polytope. Related conjectures are formulated for tight polyhedral submanifolds and generalized Heawood inequalities, including an Upper Bound Conjecture for combinatorial manifolds.

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Universität Duisburg, 47048, Duisburg, Germany

    Wolfgang Kühnel

Authors
  1. Wolfgang Kühnel
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada

    T. Bisztriczky

  2. Department of Mathematics, University College, London, UK

    P. McMullen

  3. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

    R. Schneider

  4. Department of Mathematics and Statistics, York University, North York, Ontario, Canada

    A. Ivić Weiss

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© 1994 Springer Science+Business Media Dordrecht

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Kühnel, W. (1994). Manifolds in the Skeletons of Convex Polytopes, Tightness, and Generalized Heawood Inequalities. In: Bisztriczky, T., McMullen, P., Schneider, R., Weiss, A.I. (eds) Polytopes: Abstract, Convex and Computational. NATO ASI Series, vol 440. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0924-6_11

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  • DOI: https://doi.org/10.1007/978-94-011-0924-6_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4398-4

  • Online ISBN: 978-94-011-0924-6

  • eBook Packages: Springer Book Archive

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