surface Surface 2006-06-22 14:35:01 2007-08-29 14:51:50 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 A \emph{surface} is a two-dimensional topological manifold. A closed surface is a surface without boundary. A result called the ``classification theorem'' gives us a symbolic semantics, matching the geometrical view point, in terms of genera, orientability and number of boundary components. Together with the connected sum operation, they make available a powerful language to be explored and exploited. As an example of a surface take $T=S^1\times S^1$ the two torus, the boundary of a solid sugar donut shaped cake $D^2\times S^1$, where $S^1$ is the familiar modulus one complex numbers.