A Combinatorial Analogue of Poincaré's Duality Theorem
Victor L. Klee · Canadian Journal of Mathematics · 1964
For a non-negative integer s and a finite simplicial complexK, letβS(K) denote thes-dimensional Betti number ofKand letfs(K) denote the number ofs-simplices ofK. Our theorem, like Poincaré's, applies to combinatorial manifoldsM, but it concerns the numbersfs(M) instead of the numbersβS(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 ofn-dimensional convex poly topes which have a given numberiof (n— 1)-faces. This amounts to estimating the size of the computation problem which may be involved in solving a system ofilinear inequalities innvariables, and was the original motivation for our study.