SimplicialApproximation 2007-04-08 11:42:11 2007-04-08 11:58:14 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $K$ and $L$ be simplicial complexes and $f: |K| \to |L|$ be a continuous function. A simplicial mapping $g: |K| \to |L|$ which is homotopic to $f$ is called a \emph{simplicial approximation} of $f$. For example, suppose that $L$ is the closure of an $n$-simplex and $a_0$ is a vertex of $L$. Let $f$ be a continuous map of $|K|$ to $|L|$ where $K$ is some simplicial complex. Then the map $g$ that sends all of $K$ to $a_0$ is a simplicial approximation of $f$.