论文标题
厚的同位素属性和Heegaard分裂的映射课程组
Thick isotopy property and the mapping class groups of Heegaard splittings
论文作者
论文摘要
我们为有限生成的$ 3 $ manifold的Heegaard分裂空间的基本组提供了必要的条件。这种情况正是通过冷，Gabai和Ketover证明的厚的同位素引理的结论，该结论说，Heegaard表面的任何同位素都是由$ 1 $ $ $的表面的参数家族来实现的，面积在上方由通用常数界面，并且具有“厚度属性”。我们还证明，如果在拓扑上很小（在巴赫曼的意义上），则双曲线或球形$ 3 $ manifold的远程分裂可以满足该条件，并且其磁盘复合体具有有限生成的同型群体。总之，这样的Heegaard分裂已经有限地生成的映射类小组。
We give a necessary and sufficient condition for the fundamental group of the space of Heegaard splittings of an irreducible $3$-manifold to be finitely generated. The condition is exactly the conclusion of the thick isotopy lemma proved by Colding, Gabai and Ketover, which says that any isotopy of a Heegaard surface is achieved by a $1$-parameter family of surfaces with area bounded above by a universal constant and with some ``thickness property''. We also prove that a Heegaard splitting of a hyperbolic or spherical $3$-manifold satisfies the condition if it is topologically minimal (in the sense of Bachman) and its disk complex has finitely generated homotopy group. In conclusion, such a Heegaard splitting has finitely generated mapping class group.
