Dehn twists on nonorientable surfaces

Details Dehn twists on nonorientable surfaces Dehn twists on nonorientable surfaces

2006-01-07

arxiv.org/abs/math/0601137v1

Mathematics

Geometric Topology

33 pages, 28 figures, to appear in Fundamenta Mathematicae

Scientific paper

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(,) 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.

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