Automorphisms of the mapping class group of a nonorientable surface

Abstract

Let S be a nonorientable surface of genus g ≥ 5 with n ≥ 0 punctures, and Mod(S) its mapping class group. We define the complexity of S to be the maximum rank of a free abelian subgroup of Mod(S). Suppose that S 1 and S 2 are two such surfaces of the same complexity. We prove that every isomorphism Mod(S 1 ) → Mod(S 2 ) is induced by a diffeomorphism S 1 → S 2 . This is an analogue of Ivanov's theorem on automorphisms of the mapping class groups of an orientable surface, and also an extension and improvement of the first author's previous result.

Injective homomorphisms of mapping class groups of non-orientable surfaces

“…We turn now to prove an algebraic characterization of the Dehn twists along the elements of T 0 (N), up to roots and powers. This result and its proof are independent from the characterizations given in Atalan [1,2] and Atalan-Szepietowski [3], and they are interesting by themselves.…”

“…This characterization seems interesting by itself and we believe that it could be used for other purposes. Note also that this characterization is independent from related works of Atalan [1,2] and Atalan-Szepietowski [3].…”

A Note on Chains and Bounding Pairs of Dehn Twists

“…C M(N) (t a 2 ); (2) In the even genus case, the twists t a 1 and t a Indeed, in [2], it has been shown that any automorphism of the mapping class group is induced by a homeomorphism of the surface and therefore any chain of Dehn twists bounding a disc or a bounding pairs of Dehn twists are mapped to elements of this type by any automorphism of the mapping class group. Our results provide independent proofs of these facts.…”

“…Algebraic characterizations of elements of the mapping class groups and their subgroups play important role in low-dimension al topology. One can find such studies in [1,2,6,7,10], and [5]. In this note, we will concentrate on Dehn twists about two-sided simple closed curves with nonorientable complements, on nonorientable surfaces.…”

“…We know an algebraic characterization for Dehn twists about nonseparating simple closed curves with nonorientable complements (see[1,2]). Therefore, by the paragraph in Subsection 2.2, any automorphism : M(N) → M(N) maps a chain of Dehn twists of length at least two to another chain of Dehn twists of the same length.…”