Equivariant Floer Homology, Concordance, and Homology Cobordism

这个项目将研究拓扑学和几何现象，使用来自现代数学几个部分的工具。这个项目的激励问题是关于四维几何形状必须满足哪些约束。解决这些问题的主要技术工具，数学规范理论，起源于物理学，本项目的一部分将是关于进一步发展数学规范理论的技术。另一个目标是支持研究生教育，并传播数学成果。更技术性的是，该项目将研究与结和三流形相关的Floer和Khovanov理论，并推断拓扑应用。在Floer理论中，最近对三流形的Floer同调进行了一些强有力的增强。一方面，Hendricks-Manolescu定义了对合Floer同调，另一方面，Seiberg-Witten Floer谱也有了一些新的进展。这些新构造为弗洛尔同调群的结构和应用提供了新的见解，但这些新理论在实践中特别难以计算。这个项目的中心是发展我们对这些不变量和相关不变量的理解，并将它们应用于几何和拓扑问题，特别是在同调协同上。主要目标包括证明Seiberg-Witten Floer同伦理论中的手术精确三角形以及确定Seiberg-Witten Floer空间的同伦类型。此外，研究人员还提出了开发计算对合花同调的技术，以期在3流形的同调配群中发现新的现象。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。

This project will study phenomena in topology and geometry, using tools from several parts of modern mathematics. The motivating questions for this project are about what constraints a four-dimensional geometric shape must satisfy. The main technical tool to approach such questions, mathematical gauge theory, originates from physics, and part of this project will be about further developing techniques in mathematical gauge theory. An additional goal is to support graduate education, and to disseminate mathematical results.More technically, this project will investigate Floer and Khovanov theories associated to knots and three-manifolds, and deduce topological applications. In Floer theory, there have recently been several enhancements of the powerful and well-studied Floer homologies of three-manifolds. On one hand, Hendricks-Manolescu have defined involutive Floer homology, and on the other there have been several recent developments for Seiberg-Witten Floer spectra. These new constructions have provided new insight into the structure and applications of Floer homology groups, but these new theories have been particularly difficult to compute, in practice. This project is centered on developing our understanding of these and related invariants, and applying them to questions in geometry and topology, especially on homology cobordism. Primary goals include proving surgery exact triangles in Seiberg-Witten Floer homotopy theory as well as determining the Seiberg-Witten Floer homotopy type of Seifert spaces. In addition, the principal investigator proposes to develop techniques for calculating involutive Floer homology, in order to find new phenomena in the homology cobordism group of 3-manifolds.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.