Equivariant Floer Homology, Concordance, and Homology Cobordism

等变 Floer 同源性、一致性和同源协调性

• 批准号: 2203828
• 负责人: Matthew Stoffregen
• 金额: $ 20万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203828&HistoricalAwards=false
• 关键词: Equivariant; Floer; Homology; Concordance; Cobordism

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.

该项目将使用现代数学多个部分的工具来研究拓扑和几何中的现象。 该项目的激励问题是四维几何形状必须满足哪些约束。 解决此类问题的主要技术工具——数学规范理论，源于物理学，该项目的一部分将是进一步开发数学规范理论技术。 另一个目标是支持研究生教育，并传播数学成果。从技术上讲，该项目将研究与结和三流形相关的 Floer 和 Khovanov 理论，并推导出拓扑应用。 在 Floer 理论中，最近对三流形的强大且经过深入研究的 Floer 同调进行了一些增强。 一方面，Hendricks-Manolescu 定义了内合 Floer 同调，另一方面 Seiberg-Witten Floer 谱最近有了一些进展。 这些新结构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。 该项目的重点是发展我们对这些和相关不变量的理解，并将它们应用于几何和拓扑问题，特别是同调配边。 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 同伦类型的 Seifert 空间。 此外，主要研究者提出开发计算内合Floer同调的技术，以发现3流形同调配边群中的新现象。该奖项反映了NSF的法定使命，并通过使用基金会的评估进行评估，认为值得支持。智力价值和更广泛的影响审查标准。