Dehn twists on nonorientable surfaces

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 an(b), b) = |n|I(a, b) 2, where I(·, ·) 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 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 M(N) generated by the twists is equal to the centre of M(N] and is generated by twists about circles isotopic to boundary components of N.