Geometric structure and Floer theory of three-dimensional manifolds

低维拓扑继续借鉴了 Floer 的开创性思想。这产生了一个充满活力的子学科；新的结构正在被揭示，难题正在被解决。这项活动与几何拓扑方面的里程碑式成就同时进行 - 例如阿戈尔和怀斯对虚拟哈肯猜想的解决——凸显了瑟斯顿计划的成功。我的建议定位于这些活动领域的联系，着眼于在瑟斯顿式几何拓扑和低维弗洛尔理论不变量之间架起桥梁。******这项研究将利用基本群，致力于揭示左序群、拉紧叶状结构和弗洛尔同源性之间的相互作用。这些结构之间的相互作用产生了大量的新研究。现在为图流形建立了推测的联系（参见我与 Hanselman、Rasmussen 和 Rasmussen 的合作）。这使用了来自有界 Floer 同调的新颖代数工具，这是 Heegaard Floer 同调的一种变体，适用于具有边界的流形。我的目标是利用这些工具来研究双曲结构在 Floer 理论中的作用。 ******理解 Floer 理论的几何基础建立在 Ozsváth-Szabó 与基本群关系相关的问题之上。我与博耶和戈登的合作提出了一种推测性的联系，这种联系一直是针对这个问题的新研究活动的催化剂。 Agol 在他的凡勃伦引文中重申了这一点，在几何三流形拓扑和弗洛尔同调之间建立联系的重要性。 ******我与 Hanselman 和 Rasmussen 的合作根据刺穿环面中的浸没曲线重新铸造了具有环面边界的流形的边界不变量。虽然这需要对所讨论的 3 流形进行温和的假设，但我们的工作与 Haiden-Katzarkov-Kontsevitch 在 Fukaya 表面类别上的工作一致。我们持续的研究旨在解释我们环境中同调镜像对称的这一进展，以便在低维方面建立新的结果。这项工作指出了可有序群和叶状理论中的有趣结构。******在相关的脉络中，推测不存在双曲整数同调球 L 空间（具有最简单可能的 Heegaard Floer 同调的流形）。这是一个需要理解双曲 3 流形和 Floer 理论之间关系的关键实例。我建议通过将曲面的映射类组作为中介对象来解决这个问题：有界 Floer 同调中的双模提供了映射类组的忠实分类表示，而双曲几何中的几何限制建议搜索与迭代映射类相关的 3 流形的稳定属性。这提出了新的代数结构，以及一个旨在理解双曲 L 空间整数同调球的缺乏性的程序。

Low-dimensional topology continues to draw on groundbreaking ideas of Floer. This has generated a vibrant sub-discipline; new structure is being uncovered and difficult problems are being solved. This activity comes alongside landmark achievements in geometric topology - e.g. Agol and Wise's resolution of the virtual Haken conjecture - highlighting successes of Thurston's program. My proposal is positioned at the nexus of these areas of activity, with a view to bridging between Thurston-style geometric topology and Floer-theoretic invariants in low-dimensions.******This research will draw on the fundamental group, working to uncover the interplay between left-orderable groups, taut foliations, and Floer homology. Interplay between these structures has generated a wealth of new research; the conjectured connection is now established for graph manifolds (see my work with Hanselman, Rasmussen and Rasmussen). This uses novel algebraic tools from bordered Floer homology, a variant of Heegaard Floer homology adapted to manifolds with boundary. I aim to bring these tools to bear on the role of hyperbolic structures in Floer theory.******Understanding the geometric underpinnings of Floer theory builds on questions of Ozsváth-Szabó pertaining to relationships with the fundamental group. My work with Boyer and Gordon formulates a conjectural connection that has been a catalyst for new research activity on this problem. Reiterating this, the importance of making connections between geometric 3-manifold topology and Floer homology was singled out by Agol in his Veblen citation. ******My work with Hanselman and Rasmussen recasts bordered invariants for manifolds with torus boundary in terms of immersed curves in the punctured torus. While this requires a mild hypothesis on the 3-manifold in question, our work aligns with that of Haiden-Katzarkov-Kontsevitch on Fukaya categories of surfaces. Our continued research aims to interpret this progress in homological mirror symmetry in our setting in order to establish new results in low-dimensions. This work points to interesting structure both for orderable groups and in foliation theory.******In a related vein, it is conjectured that there do not exist hyperbolic integer homology sphere L-spaces (manifolds with simplest possible Heegaard Floer homology). This is a key instance where an understanding of the relationship between hyperbolic 3-manifolds and Floer theory is required. I propose to approach this problem with mapping class groups of surfaces as a mediating object: bimodules in bordered Floer homology provide a faithful categorical representation of the mapping class group, while geometric limits in hyperbolic geometry suggest a search for stable properties of 3-manifolds associated with iterated mapping classes. This suggests new algebraic structures, and a program towards understanding the paucity of hyperbolic L-space integer homology spheres.