2003/10/30 by Ethan D. Bloch, Bloch, Ethan D.
Computer Science · Mathematics · #52B70 (Primary) #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.CO #math.GT #msc:52B70
paper · pdf · doi:10.48550/arxiv.math/0310473
19 pages, 2 figures, Latex
arxiv created 2003/10/30 · openalex publication_date 2003/10/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 the Euler characteristic of the link of each i-simplex equals 2, where i is a non-negative even integer that is less than n. As a corollary, an elementary proof is given that any such simplicial complex has Euler characteristic zero.