Dehn twists on nonorientable surfaces

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.Comment: 33 pages, 28 figures, to appear in Fundamenta Mathematica

The Birman exact sequence for 3-manifolds

We study the Birman exact sequence for compact $3$--manifolds, obtaining a complete picture of the relationship between the mapping class group of the manifold and the mapping class group of the submanifold obtained by deleting an interior point. This covers both orientable manifolds and non-orientable ones.Comment: 30 pages, no figures. v2: Major re-write following referee suggestions. To appear in Bull. Lond. Math. Soc.; v1: This paper gives an alternative, more algebraic, proof of the main result of arXiv:1310.7884 (with less exposition

Unstable maps

A map which is non-orientable or has non-empty boundary has a canonical double cover which is orientable and has empty boundary. The map is called stable if every automorphism of this cover is a lift of an automorphism of the map. This note describes several infinite families of unstable maps, and relates them to similar phenomena for graphs, hypermaps and Klein surfaces.Comment: 11 pages, 4 figure