Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Abstract

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 $N$.