nLab
polyhedron

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

The term Polyhedron is used in various senses in topology and geometry. Geometrically one has the following:

Definition(Geometric)

A polyhedron is a 3-dimensional polytope.

An alternative use of the term is

‘Polyhedra = polyhedral spaces’

In classical algebraic topology, (e.g. in Spanier’s book), there is another use for the term, more or less as an abbreviation for ‘polyhedral space’. In this sense, it is given by

Definition

A space XX is called a polyhedron if it is homeomorphic to the geometric realisation of a simplicial complex and hence has a triangulation.

The classical study of polyhedra in this second sense has been one of the sources for methods and applications in algebraic topology, and it is often useful to go back to see the motivations and applications in the older sources. For instance, shape morphisms between polyhedral spaces are just homotopy classes of continuous maps so the Cech invariants of polyhedra coincide with their ordinary ‘standard’ invariants.

References

(The link gives an up-to-date bibtex reference to the more recent edition of this.)

Revised on July 17, 2013 04:10:29 by Urs Schreiber (82.169.65.155)