Dehn twists on nonorientable surfaces

Michał Stukow¹

¹ Institute of Mathematics University of Gdańsk Wita Stwosza 57 80-952 Gdańsk, Poland

Fundamenta Mathematicae, Tome 189 (2006) no. 2, pp. 117-147

Résumé

Let $t_a$ be the Dehn twist about a circle $a$ on an orientable surface. It is well known that for each circle $b$ and an integer $n$, $I(t_a^n(b),b)=|n|I(a,b)^2$, where $I(\cdot,\cdot)$ is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties of twists on nonorientable surfaces. We also prove that if ${\cal M}(N)$ is the mapping class group of a nonorientable surface $N$, then up to a finite number of exceptions, the centraliser of the subgroup of ${\cal M}(N)$ generated by the twists is equal to the centre of ${\cal M}(N)$ and is generated by twists about circles isotopic to boundary components of $N$.

Détail

DOI : 10.4064/fm189-2-3

Keywords: dehn twist about circle orientable surface known each circle integer n b where cdot cdot geometric intersection number prove similar formula circles nonorientable surfaces corollary prove algebraic properties twists nonorientable surfaces prove cal mapping class group nonorientable surface finite number exceptions centraliser subgroup cal generated twists equal centre cal generated twists about circles isotopic boundary components nbsp