Applications of Klee's Dehn-Sommerville relations

2008/05/19 by Isabella Novik, Novik, Isabella, Ed Swartz +1 · 2 citations
Chemistry · Mathematics · #13F55 #52B05 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Molecular spectroscopy and chirality

paper · pdf · doi:10.48550/arxiv.0805.2773

openalex publication_date 2008/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Triangulated Manifolds with Few Vertices: Combinatorial Manifolds

Cited by

• The Euler characteristic of a triangulated manifold in terms of even-dimensional faces
• Dehn Sommerville Manifolds

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