Geometric structure and Floer theory of three-dimensional manifolds

低维拓扑继续借鉴Floer的开创性思想。这产生了一个充满活力的分支学科；新的结构正在被发现，难题正在得到解决。这项活动伴随着几何拓扑学的里程碑式的成就--例如，Agol和Wise对虚拟Haken猜想的解决--突出了瑟斯顿计划的成功。我的建议定位在这些活动区域的结点上，以期在瑟斯顿式几何拓扑和低维Floer理论不变量之间架起桥梁。这项研究将利用基本群，努力揭示左可序群、紧叶子和Floer同调之间的相互作用。这些结构之间的相互作用产生了大量新的研究；猜测的联系现在被建立在图流形上(参见我与Hanselman、Rasmussen和Rasmussen的工作)。这使用了来自边界Floer同调的新的代数工具，它是Heegaard Floer同调的变体，适用于具有边界的流形。我的目标是将这些工具应用于双曲线结构在弗洛尔理论中的作用。理解弗洛尔理论的几何基础建立在Ozsvth-Szab关于与基本群的关系的问题上。我与博耶和戈登的研究形成了一种猜想联系，这一直是关于这个问题的新研究活动的催化剂。重申这一点，Agol在他的Veblen引文中特别指出了几何3-流形拓扑和Floer同调之间的联系的重要性。我与Hanselman和Rasmussen的工作重塑了具有环面边界的流形的边界不变量，即被穿透的环面中的浸入曲线。虽然这需要在所讨论的3-流形上作温和的假设，但我们的工作与Hayden-Katzarkov-Kontsevitch关于Fukaya曲面范畴的工作是一致的。我们继续研究的目的是在我们的背景下解释同源镜像对称性的这一进展，以便在低维领域建立新的结果。这项工作指出了可序群和叶化理论的有趣结构。在相关的脉络中，推测不存在双曲整数同调球面L-空间(具有最简单可能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。我建议将曲面的映射类群作为中介对象来处理这个问题：有界Floer同调中的双模提供了映射类群的忠实范畴表示，而双曲几何中的几何极限则建议寻找与迭代映射类相关的3-流形的稳定性质。这提出了新的代数结构，并为理解双曲L-空间整数同调球面的稀缺性提供了一个程序。

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 Ozsvth-Szab 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.