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The Angle Defect for Odd-Dimensional Simplicial Manifolds

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Abstract

In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a direct generalization of the angle defect. The generalized angle defect is not zero at the simplices of every odd-dimensional manifold. In this paper we use a sequence based upon the Bernoulli numbers to define a variant of the angle defect for finite simplicial complexes that still satisfies a Gauss-Bonnet-type theorem, but is also zero at any simplex of an odd-dimensional simplicial complex K (of dimension at least 3), such that χ(link(ηi, K)) = 2 for all i-simplices ηi of K, where i is an even integer such that 0 ≤ i ≤ n – 1. As a corollary, an elementary proof is given that any such simplicial complex has Euler characteristic zero.

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Correspondence to Ethan D. Bloch.

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Bloch, E. The Angle Defect for Odd-Dimensional Simplicial Manifolds. Discrete Comput Geom 35, 311–328 (2006). https://doi.org/10.1007/s00454-005-1221-z

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Keywords

  • Computational Mathematic
  • Simplicial Complex
  • Euler Characteristic
  • Characteristic Zero
  • Elementary Proof