It is well known that the Euler characteristic of an odd dimensional compact
manifold is zero. An Euler complex is a combinatorial analogue of a compact
manifold. We present here an elementary proof of the corresponding result for
Euler complexes

MacLaurin, Colin

Robertson, Guyan

English

arXiv

Abstract. It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes. 1. Background We begin by recalling background information on simplicial complexes and the motivating result from topology. Only finite simplicial complexes will be considered, and we include this condition in the definitions. Geometric Simplicial Complexes. Let S be a finite subset of Rn. An affine combination of the points pi ∈ S is a point x = λipi, where λi ∈ R, λi = 1. The set S is called affinely independent if no point in S is an affine combination of the other points. A convex combination is an affine combination with non-negative coefficients λi. The convex hull, convS is the set of all convex combinations of the points in S. A (geometric) k-simplex is the convex hull of a set of k + 1 affinely independent points in Rn. We say k is the dimension of the simplex. In Rn there are simplices ranging from dimension −1 through to n. Let S ⊂ Rn be finite and affinely independent (so σ = convS is a simplex). If T ⊂ S then T is also affinely independent, so the convex hull τ = conv T is a simplex as well. We say τ is a face of σ. Example. A 2-simplex in R2 is a triangle, and has eight faces: the triangle itself, its three edges, three vertices, and also the empty set ∅. A (geometric) simplicial complex is a finite collection C of simplices such that (i) if σ ∈ C, and τ is a face of σ then τ ∈ C; (ii) if σ, τ ∈ C then σ ∩ τ is a face of both σ and τ. Note that σ and τ may be be disjoint in (ii), since ∅ is a face of every simplex. The dimension of C is the highest dimension of a simplex contained in C. Let C be an n-dimensional simplicial complex. The polyhedron or underlying space of C, written |C|, is the topological space formed by taking the subset of Rn consisting of points which are covered by simplices in C, with the usual topology inherited fro

Euler Characteristic in Odd Dimensions

Abstract

Similar works

Full text

Available Versions

CiteSeerX