Dehn twists on nonorientable surfaces

Volume 189 / 2006

Michał Stukow Fundamenta Mathematicae 189 (2006), 117-147 MSC: Primary 57N05; Secondary 20F38, 57M99. DOI: 10.4064/fm189-2-3


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 $N$.


  • Michał StukowInstitute of Mathematics
    University of Gdańsk
    Wita Stwosza 57
    80-952 Gdańsk, Poland

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