Automorphisms of the mapping class group of a nonorientable surface

An algebraic characterization of a Dehn twist for nonorientable surfaces

“…We note that Dehn twists about two-sided nonseparating simple closed curves with nonorientable complements is already known ( [2] or [1]). (See Theorem 2.1 in [4] for characterization for Dehn twists about nonseparating simple closed curves on orientable surfaces.)…”

“…As mentioned in the introduction, algebraic characterizations of Dehn twists about nonseparating simple closed curves curves in both orientable (Theorem 2.1. of [4]) and nonorientable surfaces (provided that the curve is not characteristic, Theorem 3.1. of [1]; see also [2];) are already done. On the other hand, the situation for separating curves is more involved.…”

“…Let a 1 , a 2 , a 3 be distinct, nonisotopic, nonseparating two-sided simple closed curves with nonorientable complements. We say that they form a triangle if each geometric intersection i(a i , a j ) = 1, for all i = j (see [2]). A triangle is called orientation reversing if there are Dehn twists about these curves, denoted by t a 1 , t a 2 and t a 3 , so that…”

“…An algebraic characterization of a Dehn twist plays important role in the computation of the outer automorphism group of the mapping class group of a surface (orientable or nonorientable, see [1], [2] and [4]). Moreover, it is one of the main tools in the proof of the fact that any injective endomorphism of the mapping class group of an orientable surface must be an isomorphism proved by Ivanov and McCarthy ([6]).…”

“…Therefore, the algebraic characterization about separating curves in [1] has also gaps. In [2], using different techniques, we gave an algebraic characterization for Dehn twists about nonseparating simple closed curves with nonorientable complements. However, an algebraic characterization of the Dehn twists about other two-sided simple closed curves is still missing.…”