Reducible mapping class of the canonical Heegaard splitting in a mapping torus
Reducible mapping class of the canonical Heegaard splitting in a mapping torus
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DOI：
10.1016/j.topol.2019.03.015
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发表时间：
2019-05
影响因子：
0.6
通讯作者：
Faze Zhang;Y. Zou
中科院分区：
文献类型：
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作者：
Faze Zhang;Y. Zou
设S是亏格至少为2的可定向闭曲面。从S× I出发，对于给定的从S×{1}到S×{0}的方向反转同胚f，存在一个可定向的闭3-流形Mf = S× I/f，称之为映射环面.已知M f允许正则Heegaard分裂H1 H2.通过Namazi [H. Namazi，Topology Appl.154（2007），no.16，2939-2949]，该Heegaard分裂的映射类群，记为M od（H; H1，H2），包含一个具有无穷阶的可约映射类.因此，对于给定的Mod（H; H1，H2）中的元素，知道它是否是可约的是很有趣的.利用f在曲线复形中的平移长度，证明了若f是恒等映射或其平移长度至少为8，则Mod（H; H1，H2）的每个元素都是可约的.
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 M f= S× I/f which is called a mapping torus. It is known that M f admits a canonical Heegaard splitting H 1∪ Σ 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 M o d (Σ; 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 M o d (Σ; 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 M o d (Σ; H 1, H 2) is reducible.