EulerrCharacteristic 2006-09-02 16:01:39 2007-05-24 18:12:33 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 The term \emph{Euler characteristic} is defined for several objects. If $K$ is a finite simplicial complex of dimension $m$, let $\alpha_i$ be the number of simplexes of dimension $i$. The \emph{Euler characteristic} of $K$ is defined to be $$ \chi(K) = \sum_{i=0}^m (-1)^i \alpha_i . $$ Next, if $K$ is a finite CW complex, let $\alpha_i$ be the number of i-cells in $K$. The \emph{Euler characteristic} of $K$ is defined to be $$ \chi(K) = \sum_{i \ge 0}(-1)^i \alpha_i . $$ If $X$ is a finite polyhedron, with triangulation $K$, a simplicial complex, then the \emph{Euler characteristic} of $X$ is $\chi(K)$. It can be shown that all triangulations of $X$ have the same value for $\chi(K)$ so that this is well-defined. Finally, if $C=\{C_q\}$ is a finitely generated graded group, then the \emph{Euler characteristic} of $C$ is defined to be $$ \chi(C) = \sum_{q \ge 0} (-1)^q rank(C_q) . $$