A Combinatorial Analogue of Poincaré's Duality Theorem

Abstract

For a non-negative integer s and a finite simplicial complex K, let β_S(K) denote the s-dimensional Betti number of K and let f_s(K) denote the number of s-simplices of K. Our theorem, like Poincaré's, applies to combinatorial manifolds M, but it concerns the numbers f_s(M) instead of the numbers β_S(M). One of the formulae given below is used by the author in (5) to establish a sharp upper bound for the number of vertices of n-dimensional convex poly topes which have a given number i of (n — 1)-faces. This amounts to estimating the size of the computation problem which may be involved in solving a system of i linear inequalities in n variables, and was the original motivation for our study.