Topology of 4-Manifolds, Embeddings, and Stable Homotopy Invariants of Links

这个项目涉及拓扑形状的研究，或流形，在维度3和4。在欧几里得空间上局部建模的这些形状的分类在这些维度上是一个重要的问题，原因如下。其一是它与物理学的相关性；事实上，理论物理学的思想已经导致了对这些维度空间结构的新见解。此外，拓扑学中一些突出的开放问题特别关注4维流形，对这类流形的研究与许多数学领域有着重要的联系。本课题研究了4流形拓扑中的几个问题，包括具有大基本群的流形的连续变形分类，其中许多环不能收缩。该项目还具有实质性的更广泛的影响，旨在培训本科生和研究生，并开展外展活动。本课题针对4流形中连杆不变量、连杆一致性不变量、曲面分类等常见问题，提出了不同的研究方法和应用。一个目标是建立在最近的进展，以解决拓扑手术猜想，一个开放的问题，在4流形拓扑，自由基本群。本工作将涉及通用外科问题的构建，以及A-B切片问题的相关方法。该项目还旨在将函数的Goodwillie-Weiss嵌入微积分技术应用于4流形拓扑中的嵌入问题，特别是链接一致性和4流形中的曲面嵌入问题。另一个研究方向是连杆同伦理论的稳定同伦细化，以及在4流形曲面上的应用。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 concerns the study of topological shapes, or manifolds, in dimensions 3 and 4. The classification of such shapes, locally modeled on the Euclidean space, in these dimensions is an important problem for several reasons. One is its relevance to physics; indeed, ideas from theoretical physics have led to new insights into the structure of spaces of these dimensions. Moreover, some of the outstanding open problems in topology concern manifolds specifically of dimension 4, and the study of such manifolds has important links with many areas of mathematics. This project is aimed at several questions in 4-manifold topology, including the classification up to continuous deformations of manifolds with large fundamental groups where many loops cannot be contracted. The project also has substantial broader impacts, aimed at training undergraduate and graduate students, and outreach activities.This project is aimed at several research directions, providing different methods and applications to a common set of problems: invariants of links and of link concordance, and classification of surfaces in 4-manifolds. One goal is to build on recent advances towards a resolution of the topological surgery conjecture, an open problem in 4-manifold topology, for free fundamental groups. This work will involve the construction of universal surgery problems, and related methods of the A-B slice problem. The project also aims to apply the techniques of the Goodwillie-Weiss embedding calculus of functors to embedding problems in 4-manifold topology, in particular to link concordance and to embedding of surfaces in 4-manifolds. Another direction of research concerns stable homotopy refinement of link homology theories, with applications to surfaces in 4-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.