Applications of Klee’s Dehn–Sommerville Relations

Authors:
Isabella Novik;Ed Swartz
Affiliations:
University of Washington, Department of Mathematics, P.O. Box 354350, 98195-4350, Seattle, WA, USA;Cornell University, Department of Mathematics, 14853-4201, Ithaca, NY, USA
Venue:
Discrete & Computational Geometry - Special Issue Dedicated to the Memory of Victor Klee
Year:
2009

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Abstract

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges.