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

摘要

Let S be an orientable closed surface with genus at least two. From S x I, for a given orientation-reversing homeomorphism f from S x {1} to S x {0}, there is an orientable closed 3-manifold M-f = S x I/f which is called a mapping torus. It is known that Mf admits a canonical Heegaard splitting H-2 U-Sigma H-2. 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(Sigma; H-1, H-2), contains a reducible mapping class which has infinitely order. So it is interesting to know that for a given element in Mod(Sigma; H-1, H-2), 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(Sigma; H-1, H-2) is reducible. (C) 2019 Elsevier B.V. All rights reserved.