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).

中文翻译：

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定量 Heegaard Floer 上同调和 Calabi 不变量

我们定义了一个新的光谱不变量族，该族与任何属的紧凑和连接曲面中的某些拉格朗日链接相关联。我们表明我们的不变量在其极限内恢复了哈密顿量的卡拉比不变量。作为应用，我们解决了拓扑曲面动力学和连续辛拓扑的几个开放性问题：我们将Calabi同态扩展到Oh和Müller构造的同胚群，并在双球面的面积和方向保持同胚群上构造了一个无限维的拟态射族。我们的不变量的灵感来自 Polterovich 和 Shelukhin 最近的工作，通过 orbifold Floer 同源性定义和应用光谱不变量，对于由双球体中的平行圆组成的链接。我们工作的一个特点是它避免了 orbifold 设置，而是依赖于“经典”Floer 同源性。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。