Proof of the simplicity conjecture
Proof of the simplicity conjecture
复制标题
简单性猜想的证明
DOI：
10.4007/annals.2024.199.1.3
复制
发表时间：
2020
影响因子：
4.9
通讯作者：
Sobhan Seyfaddini
中科院分区：
文献类型：
--
作者：
Daniel Cristofaro;Vincent Humilière;Sobhan Seyfaddini
20 世纪 70 年代，Fathi 证明了 $n$ 球的紧支撑体积保持同胚群对于 $n \ge 3$ 来说是简单的，并询问相同的陈述在维度 $2$ 中是否成立。我们证明了二圆盘的紧支撑保面积同胚群并不简单。这证实了所谓的“简单性猜想”。事实上，我们证明了先验更强的陈述，即该群并不完美。
我们证明中的一个重要步骤涉及验证 Hutchings 对某些平滑扭曲映射的猜想，该猜想涉及从使用周期弗洛尔同调定义的谱不变量的渐近恢复卡拉比不变量。另一个关键步骤基于连续辛拓扑的最新进展，包括证明这些谱不变量连续延伸到圆盘的面积保持同胚。 PFH 光谱不变量的这两个属性可能具有独立的意义。
我们的总体策略部分受到Fathi的建议和Oh针对简单性问题的方法的启发。特别是，我们证明了 Oh 研究的无限扭曲映射不是有限能量同胚，这肯定地解决了“无限扭曲猜想”；这些扭曲图现在是哈密顿同胚的第一个例子，可以说它具有无限的能量。我们工作的另一个结果是，适用于高维球的体积保持同胚的各种形式的碎片在二维中失败。
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.