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LOWER BOUND THEOREMS AND A GENERALIZED LOWER BOUND CONJECTURE FOR BALANCED SIMPLICIAL COMPLEXES

Part of: Polytopes and polyhedra Algebraic combinatorics PL-topology

Published online by Cambridge University Press:  01 February 2016

Steven Klee  and
Isabella Novik
Steven Klee
Affiliation:
Department of Mathematics, Seattle University, Seattle, WA 98122, U.S.A. email klees@seattleu.edu
Isabella Novik
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A. email novik@math.washington.edu

Abstract

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A $(d-1)$ -dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$ -coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated lower bound theorem (LBT) for normal pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; and we propose the balanced analog of the generalized lower bound conjecture (GLBC) and establish some related results. We close with constructions of balanced manifolds with few vertices.

MSC classification

Primary: 05E45: Combinatorial aspects of simplicial complexes
Secondary: 52B12: Special polytopes (linear programming, centrally symmetric, etc.) 52B70: Polyhedral manifolds 57Q15: Triangulating manifolds

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 441 - 477
Copyright
Copyright © University College London 2016 

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