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A Combinatorial Analogue of Poincaré's Duality Theorem

  • Victor Klee (a1)
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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.

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References
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1. Alexandroff, Paul and Hopf, Heinz, Topologie I (Berlin, 1935).
2. Feller, William, An introduction to probability theory and its applications (New York, 1951).
3. Hadwiger, H., Eulers Charakteristik und kombinatorische Géométrie, J. Reine Angew. Math., 194 (1955), 101110.
4. Klee, Victor, The Euler characteristic in combinatorial geometry, Am. Math. Monthly, 70 (1963), 119127.
5. Klee, Victor, The number of vertices of a convex polytope; to appear in this Journal.
6. Weyl, H., Elementare Théorie der konvexen Polyeder, Comment. Math. Helv., 7 (1935), 290306.
7. Fieldhouse, M., Linear programming, Ph.D. Thesis, Cambridge Univ. (1961). [Reviewed in Operations Res. 10 (1962), 740.
8. Lefschetz, S., Introduction to topology (Princeton, 1949).
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.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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