Face Numbers of Polytopes and Complexes

Authored by: Louis J. Billera , Anders Björner

Discrete and Computational Geometry

Print publication date:  April  2004
Online publication date:  April  2004

Print ISBN: 9781584883012
eBook ISBN: 9781420035315
Adobe ISBN:

10.1201/9781420035315.ch18

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Abstract

Geometric objects are often put together from simple pieces according to certain combinatorial rules. As such, they can be described as complexes with their constituent cells, which are usually polytopes and often simplices. Many constraints of a combinatorial and topological nature govern the incidence structure of cell complexes and are therefore relevant in the analysis of geometric objects. Since these incidence structures are in most cases too complicated to be well understood, it is worthwhile to focus on simpler invariants that still say something nontrivial about their combinatorial structure. The invariants to be discussed in this chapter are the f-vectors f = (f 0, f 1, …), where fi is the number of i-dimensional cells in the complex.

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