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The Powell conjecture and reducing sphere complexes
Journal of the London Mathematical Society ( IF 1.0 ) Pub Date : 2019-07-30 , DOI: 10.1112/jlms.12272 Alexander Zupan 1
Journal of the London Mathematical Society ( IF 1.0 ) Pub Date : 2019-07-30 , DOI: 10.1112/jlms.12272 Alexander Zupan 1
Affiliation
The Powell conjecture offers a finite generating set for the genus Goeritz group, the group of automorphisms of that preserve a genus Heegaard surface , generalizing a classical result of Goeritz in the case . We study the relationship between the Powell conjecture and the reducing sphere complex , the subcomplex of the curve complex spanned by the reducing curves for the Heegaard splitting. We prove that the Powell conjecture is true if and only if is connected. Additionally, we show that reducing curves that meet in at most six points are connected by a path in ; however, we also demonstrate that even among reducing curves meeting in four points, the distance in between such curves can be arbitrarily large. We conclude with a discussion of the geometry of .
中文翻译:
鲍威尔猜想和约化球体络合物
鲍威尔猜想为该类提供了一个有限的生成集 Goeritz群,自同构群 保留一个属 Heegaard表面 ,在该案例中推广了Goeritz的经典结果 。我们研究了鲍威尔猜想和还原球体之间的关系,曲线复数的子复数 Heegaard分裂的归约曲线所覆盖。我们证明Powell猜想是当且仅当已连接。此外,我们证明了在最多六个点处相交的归约曲线是通过一条路径连接的; 但是,我们也证明了即使在四个点相交的简化曲线中,这样的曲线之间的距离可以任意大。最后,我们讨论了。
更新日期:2019-07-30
中文翻译:
鲍威尔猜想和约化球体络合物
鲍威尔猜想为该类提供了一个有限的生成集 Goeritz群,自同构群 保留一个属 Heegaard表面 ,在该案例中推广了Goeritz的经典结果 。我们研究了鲍威尔猜想和还原球体之间的关系,曲线复数的子复数 Heegaard分裂的归约曲线所覆盖。我们证明Powell猜想是当且仅当已连接。此外,我们证明了在最多六个点处相交的归约曲线是通过一条路径连接的; 但是,我们也证明了即使在四个点相交的简化曲线中,这样的曲线之间的距离可以任意大。最后,我们讨论了。