An algebraic characterization of a Dehn twist for nonorientable surfaces

Abstract

Let N k g be a nonorientable surface of genus g ≥ 5 with k-punctures. In this note, we will give an algebraic characterization of a Dehn twist about a simple closed curve on N k g . Along the way, we will fill some little gaps in the proofs of some theorems in [1] and [4] giving algebraic characterizations of Dehn twists about separating simple closed curves. Indeed, our results will give an algebraic characterization for the topological type of Dehn twists about separating simple closed curves.

Injective homomorphisms of mapping class groups of non-orientable surfaces

“…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].…”

“…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.…”

Abelian subgroups of the mapping class groups for non-orientable surfaces

“…Thus, we give another proof of this result for odd genus non-orientable surfaces whose Euler characteristic are negative. However, for even genus non-orientable surfaces, we don't know the maximal rank of them (Atalan [1,Proposition 3.1] gave a partial answer for it). We give the answer for this question.…”