Reducible mapping class of the canonical Heegaard splitting in a mapping torus

Abstract

Let S be an orientable closed surface with genus at least two. From S×I, for a given orientation-reversing homeomorphism f from S×{1} to S×{0}, there is an orientable closed 3-manifold Mf=S×I/f which is called a mapping torus. It is known that Mf admits a canonical Heegaard splitting H1∪ΣH2. By the construction of Namazi [H.Namazi, Topology Appl. 154 (2007), no. 16, 2939-2949], the mapping class group of this Heegaard splitting, denoted by Mod(Σ;H1,H2), contains a reducible mapping class which has infinitely order. So it is interesting to know that for a given element in Mod(Σ;H1,H2), whether it is reducible or not.Using the translation length of f in the curve complex, we prove that if f is the identity map or its translation length is at least 8, then each element of Mod(Σ;H1,H2) is reducible.