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.