Hostname: page-component-5d59c44645-k78ct Total loading time: 0 Render date: 2024-02-26T14:14:47.596Z Has data issue: false hasContentIssue false

A Combinatorial Analogue of Poincaré's Duality Theorem

Published online by Cambridge University Press:  20 November 2018

Victor Klee*
University of Washington and Boeing Scientific Research Laboratories
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a non-negative integer s and a finite simplicial complex K, let βS(K) denote the s-dimensional Betti number of K and let fs(K) denote the number of s-simplices of K. Our theorem, like Poincaré's, applies to combinatorial manifolds M, but it concerns the numbers fs(M) instead of the numbers βS(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.

Research Article
Copyright © Canadian Mathematical Society 1964


1. Alexandroff, Paul and Hopf, Heinz, Topologie I (Berlin, 1935).Google Scholar
2. Feller, William, An introduction to probability theory and its applications (New York, 1951).Google Scholar
3. Hadwiger, H., Eulers Charakteristik und kombinatorische Géométrie, J. Reine Angew. Math., 194 (1955), 101110.Google Scholar
4. Klee, Victor, The Euler characteristic in combinatorial geometry, Am. Math. Monthly, 70 (1963), 119127.Google Scholar
5. Klee, Victor, The number of vertices of a convex polytope; to appear in this Journal.Google Scholar
6. Weyl, H., Elementare Théorie der konvexen Polyeder, Comment. Math. Helv., 7 (1935), 290306.Google Scholar
7. Fieldhouse, M., Linear programming, Ph.D. Thesis, Cambridge Univ. (1961). [Reviewed in Operations Res. 10 (1962), 740.Google Scholar
8. Lefschetz, S., Introduction to topology (Princeton, 1949).Google Scholar
9. Sommerville, D. M. Y., The relations connecting the angle sums and volume of a polytope in space of n dimensions, Proc. Roy. Soc. London, Ser. A, 115 (1927), 103119.Google Scholar