An elementary approach to the mapping class group of a surface

Abstract

We consider an oriented surface $S$ and a cellular complex $X$ of curves on $S$, defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex $X$ is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of $S$, following the ideas of Hatcher–Thurston and Harer.