HandleDecomposition 2005-06-21 17:56:41 2005-06-24 05:38:05 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 $M^n$ be a smooth, connected, closed $n$ dimensional manifold. A {\em handle} is $H_{\lambda}^n = B^{\lambda} \times B^{n-\lambda}$ where $B^{\lambda}$ is a $\lambda$-ball. Any such manifold $M$ is diffeomorphic to the union of finitely many such handles where each handle $H_{\lambda}^n$ is in a one-to-one correspondence with the critical points of \PMlinkescapetext{index} $\lambda$ of a Morse function on $M$.