Higher Structure in Low-Dimensional Floer Theories

早在有记载的历史之前，人类就对三维空间的几何学感兴趣；自从爱因斯坦提出广义相对论以来，四维空间的几何也变得至关重要，其应用如今已像全球定位系统一样无处不在。数学家将空间研究分为两部分：定性问题（如空间中是否有洞），这是拓扑领域；定量问题（如洞的直径），这是几何领域。大多数，虽然不是全部，应用程序都需要定量的答案，但是在开始定量的答案——几何——之前，人们需要理解定性的行为——低维拓扑，这是这个国家科学基金会资助项目的重点。事实上，在四维空间中，甚至许多最基本的拓扑问题仍然没有答案——诸如有多少不同的简单空间，如何将空间切割成更简单的部分，以及在给定的四维空间中可以容纳什么样的表面等问题。通过利用与数学和高能物理其他部分的关系，本提案寻求通过开发工具来回答这些基本问题，从而促进人类的理解。本项目主要研究辛拓扑、代数拓扑和表示理论在低维拓扑中的应用，有四个相互关联的目标。第一个目标是对具有环面边界的三流形将Lipshitz-Ozsváth-Thurston的“有边界Heegaard flower同调”不变量从“hat”变体扩展到更丰富的“minus”变体。第二个目标是证明Heegaard flower同源性的更高的自然性结果，给出一个从修饰配合物的适当拟范畴到链配合物拟范畴的函子。第三个目标是证明空间的Heegaard flower同调与其分支覆盖的Heegaard flower同调之间的新的定位结果。第四是将Khovanov同伦的Lipshitz-Sarkar的稳定同伦细化推广到结点间的配合和Khovanov同伦的其他变体，包括Lee同伦。这些工具有望应用于三维和四维拓扑界面上的一致性、同源协同性和其他问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Humanity has been interested in the geometry of three-dimensional spaces since long before recorded history; since Einstein's work on general relativity, the geometry of four-dimensional spaces has also been of central importance, with applications now as ubiquitous as the global positioning system. Mathematicians have divided the study of spaces into two parts: qualitative questions (like whether a space has a hole in it), which is the field of topology, and quantitative questions (like the diameter of the hole), which is the field of geometry. Most, though not all, applications require quantitative answers, but before one can start on quantitative answers---the geometry---one needs to understand the qualitative behavior---low-dimensional topology, the focus of this National Science Foundation funded project. Indeed, in four-dimensions, even many of the most basic topological questions remain unanswered---questions about how many different simple spaces there are, how spaces can be cut up into simpler pieces, and what kinds of surfaces one can fit in given four-dimensional spaces. By using relations with other parts of mathematics and high-energy physics, this proposal seeks to advance human understanding by developing tools to answer some of these basic questions. This project, which focuses on applications of symplectic topology, algebraic topology, and representation theory to low-dimensional topology, has four interrelated goals. The first goal is to extend Lipshitz-Ozsváth-Thurston's "bordered Heegaard Floer homology" invariant of three-manifolds with boundary from the "hat" variant to the richer "minus" variant, for three-manifolds with torus boundary. The second goal is to prove a higher naturality result for Heegaard Floer homology, giving a functor from an appropriate quasi-category of decorated cobordisms to the quasi-category of chain complexes. The third goal is to prove new localization results relating the Heegaard Floer homology of a space with the Heegaard Floer homology of its branched covers. The fourth is to extend Lipshitz-Sarkar's stable homotopy refinement of Khovanov homology to cobordisms between knots and to other variants of Khovanov homology, including Lee homology. These tools are expected to have applications to concordance, homology cobordism, and other problems at the interface of three- and four-dimensional topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.