Quantitative Heegaard Floer cohomology and the Calabi invariant
Quantitative Heegaard Floer cohomology and the Calabi invariant
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DOI：
10.1017/fmp.2022.18
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发表时间：
2021-05
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影响因子：
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通讯作者：
Daniel Cristofaro-Gardiner;Vincent Humilière;C. Mak;Sobhan Seyfaddini;I. Smith
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作者：
Daniel Cristofaro-Gardiner;Vincent Humilière;C. Mak;Sobhan Seyfaddini;I. Smith
摘要我们定义了一个新的谱不变量族，它与任意亏格的紧致连通曲面中的某些拉格朗日链有关。我们表明，我们的不变量恢复的Calabi不变量的Hamilton在其极限。作为应用，我们解决了拓扑曲面动力学和连续辛拓扑学中的几个公开问题：我们证明了任何紧曲面的Hamilton同胚群（可能是空的）边界并不简单;我们将Calabi同态推广到Oh和Müller构造的同态群，并在双球面的保面积保方向同胚群上构造了一类无限维的拟态射。我们的不变量的灵感来自最近的工作Polterovich和Shelukhin定义和应用谱不变量，通过orbifold弗洛尔同源，在两个领域的平行圆组成的链接。我们的工作的一个特点是，它避免了orbifold设置，而是依赖于“经典”Floer同源。这不仅大大简化了技术背景，而且对于某些方面（例如构造拟态射的应用）似乎是必不可少的。
Abstract We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: We show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of hameomorphisms constructed by Oh and Müller, and we construct an infinite-dimensional family of quasi-morphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).