[论文解读] Proof of the simplicity conjecture

Dan Cristofaro‐Gardiner, Vincent Humilière+1 more|arXiv (Cornell University)|Jan 6, 2020

Geometric and Algebraic Topology参考文献 38被引用 11

一句话总结

该论文通过利用周期Floer同调中的谱不变量构造一个适当的正规子群，证明了2-圆盘上紧支集保面积同胚群不是单群——也不是完备群，从而解决了简洁性猜想。其关键创新在于建立了这些不变量的C⁰连续性，并验证了Hutchings关于Calabi不变量的猜想，使得Calabi同态可被延拓至哈密顿同胚。

ABSTRACT

In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n \ge 3$, asked if the same statement holds in dimension $2$. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles what is known as the "simplicity conjecture" in the affirmative. In fact, we prove the a priori stronger statement that this group is not perfect. An important step in our proof involves verifying for certain smooth twist maps a conjecture of Hutchings concerning recovering the Calabi invariant from the asymptotics of spectral invariants defined using periodic Floer homology. Another key step, which builds on recent advances in continuous symplectic topology, involves proving that these spectral invariants extend continuously to area-preserving homeomorphisms of the disc. These two properties of PFH spectral invariants are potentially of independent interest. Our general strategy is partially inspired by suggestions of Fathi and the approach of Oh towards the simplicity question. In particular, we show that infinite twist maps, studied by Oh, are not finite energy homeomorphisms, which resolves the "infinite twist conjecture" in the affirmative; these twist maps are now the first examples of Hamiltonian homeomorphisms which can be said to have infinite energy. Another consequence of our work is that various forms of fragmentation for volume preserving homeomorphisms which hold for higher dimensional balls fail in dimension two.

研究动机与目标

• 解决关于2-圆盘上紧支集保面积同胚群的长期存在的简洁性猜想。
• 通过构造一个适当的正规子群，证明该群不是完备群，从而表明其不是单群。
• 建立周期Floer同调（PFH）谱不变量在保面积同胚上的C⁰连续性。
• 验证Hutchings关于单调扭映射的PFH谱不变量渐近行为恢复Calabi不变量的猜想。
• 证明无限扭映射不是有限能量同胚，从而解决“无限扭猜想”。

提出的方法

• 使用周期Floer同调（PFH）为哈密顿微分同胚定义谱不变量，并通过C⁰连续性将其延拓至同胚。
• 利用霍弗范数和谱不变量的估计，证明PFH谱不变量在单位元处及之外均具有C⁰拓扑下的连续性。
• 为单调扭映射构建PFH的组合模型，以计算ECH指标并验证全纯曲线的正性。
• 证明对于正单调扭映射，谱不变量的渐近行为可恢复Calabi不变量，从而证实Hutchings的猜想。
• 通过证明其谱不变量发散，表明无限扭映射不是有限能量同胚，从而证明无限扭猜想。
• 利用Calabi同态的核（即一个非平凡正规子群）的存在，得出该群非单且非完备的结论。

实验结果

研究问题

• RQ1 2-圆盘上紧支集保面积同胚群是否为单群？
• RQ2 Calabi不变量能否被延拓至2-圆盘上哈密顿同胚群？
• RQ3 PFH谱不变量能否连续延拓至保面积同胚？
• RQ4 PFH谱不变量的渐近行为是否能恢复单调扭映射的Calabi不变量？
• RQ5 无限扭映射是否为有限能量同胚？

主要发现

• 群 $\text{Homeo}_c(\text{D},\text{ω})$ 不是单群，简洁性猜想得以解决。
• 群 $\text{Homeo}_c(\text{D},\text{ω})$ 不是完备群，因其存在非平凡的换位子群。
• PFH谱不变量可连续延拓至保面积同胚，确立了关键的正则性性质。
• Hutchings的猜想对正单调扭映射成立：Calabi不变量可由PFH谱不变量的渐近行为恢复。
• 无限扭映射不是有限能量同胚，证实了“无限扭猜想”。
• Calabi同态可连续延拓至哈密顿同胚群 $\text{Hameo}_c(\text{D},\text{ω})$，提供一个良定义的实值不变量。