Geometric structure and Floer theory of three-dimensional manifolds

低维拓扑继续借鉴弗洛尔的开创性思想。这产生了一个充满活力的分支学科；新格局正在显现，难题正在解决。这一活动伴随着几何拓扑学的里程碑式成就——例如Agol和Wise对虚拟哈肯猜想的解决——突出了Thurston计划的成功。我的建议定位于这些活动领域的联系，以期在瑟斯顿风格的几何拓扑和低维的花理论不变量之间架起桥梁。******这项研究将以基本群为基础，致力于揭示左序群、紧叶和花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；图流形的猜想连接现在已经建立（参见我与Hanselman， Rasmussen和Rasmussen的工作）。这使用了来自有边花同调的新代数工具，这是Heegaard花同调的一种变体，适用于有边界的流形。我的目标是把这些工具运用到弗洛尔理论中双曲结构的作用上。******理解弗洛尔理论的几何基础是建立在Ozsvth-Szab关于与基本群的关系的问题之上的。我与Boyer和Gordon的工作形成了一种推测性的联系，这种联系已经成为对这个问题进行新的研究活动的催化剂。重申这一点，在几何三流形拓扑和花同源性之间建立联系的重要性被Agol在他的Veblen引文中挑出来。******我与Hanselman和Rasmussen合作，根据刺穿环面中的浸入曲线，对具有环面边界的流形的有边不变量进行了重新定义。虽然这需要一个关于3流形的温和假设，但我们的工作与haden - katzarkov - kontsevitch关于Fukaya曲面类别的工作是一致的。我们继续研究的目的是在我们的设置中解释这一进展在同调镜像对称，以便在低维建立新的结果。这项工作指出了可序群和叶理理论的有趣结构。******在相关的情况下，我们推测不存在双曲整数同调球l -空间（具有最简单可能Heegaard花同调的流形）。这是一个关键的例子，其中双曲3流形和花理论之间的关系的理解是必要的。我建议用曲面的映射类群作为中介对象来解决这个问题：边界花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空间整数同调球的稀少性提供了一个程序。

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.