Geometric structure and Floer theory of three-dimensional manifolds

低维拓扑继续借鉴Floer的突破性思想。这产生了一个充满活力的分支学科;新的结构正在被发现，困难的问题正在被解决。这项活动伴随着几何拓扑学的里程碑式成就-例如Agol和Wise对虚哈肯猜想的解决-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Thurston风格的几何拓扑和低维Floer理论不变量之间架起桥梁。这项研究将借鉴的基本组，努力揭示左有序组，拉紧叶理，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丰富的新的研究;图流形的固定连接现在已经建立（见我与Hanselman，Rasmussen和Rasmussen的工作）。这使用了新的代数工具，从边界的弗洛尔同源，一个变种的Heegaard弗洛尔同源适应流形的边界。我的目标是利用这些工具来研究弗洛尔理论中双曲结构的作用。理解弗洛尔理论的几何基础建立在奥兹瓦特-绍博关于与基本群的关系的问题上。我与博耶和戈登的工作形成了一种精神联系，这种联系是这个问题的新研究活动的催化剂。重申这一点，使几何3-流形拓扑和弗洛尔同调之间的联系的重要性是由阿戈尔在他的凡勃伦引文挑出来。** 我与Hanselman和Rasmussen一起工作，根据穿孔环面中的浸入曲线，重新构建了具有环面边界的流形的边界不变量。虽然这需要一个温和的假设的3-流形的问题，我们的工作与Haupt-Katzarkov-Kontsevitch的福谷类别的表面。我们继续研究的目的是在我们的环境中解释同调镜像对称的这一进展，以建立低维的新结果。这项工作指出了有序群和叶理理论中有趣的结构。在一个相关的脉络中，它被证明不存在双曲整数同调球面L-空间（具有最简单可能的Heegaard Floer同调的流形）。这是一个关键的例子，需要理解双曲3流形和Floer理论之间的关系。我建议处理这个问题的映射类组的表面作为中介对象：bimodules在有边Floer同源提供了一个忠实的分类表示的映射类组，而几何限制在双曲几何建议搜索稳定的性质与迭代映射类的3-流形。这表明了新的代数结构，以及对了解双曲L-空间整数同调球的缺乏程序。

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.