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

这个项目涉及3维和4维的拓扑形状或流形的研究。由于几个原因，这些形状的分类是一个重要的问题，这些形状局部地模拟在欧几里德空间上。其一是它与物理学的关联性；事实上，理论物理学的思想导致了对这些维度的空间结构的新见解。此外，拓扑学中的一些未决问题涉及到特定的4维流形，而对这种流形的研究与许多数学领域有着重要的联系。这个项目是针对4-流形拓扑中的几个问题，包括具有大的基本群的流形的连续变形的分类，其中许多环不能收缩。该项目还具有相当广泛的影响，旨在培训本科生和研究生，以及推广活动。该项目针对几个研究方向，为一组常见问题提供不同的方法和应用：链接不变量和链接协调不变量，以及4维流形中曲面的分类。一个目标是建立在解决自由基本群的拓扑外科猜想的最新进展的基础上，拓扑外科猜想是4流形拓扑中的一个公开问题。这项工作将涉及通用外科问题的构造，以及A-B切片问题的相关方法。该项目还旨在将函子的Goodwillie-Weiss嵌入演算的技巧应用于4-流形拓扑中的嵌入问题，特别是在4-流形中的链协和和曲面的嵌入问题。另一个研究方向涉及链环同调理论的稳定同伦精化，并应用于4-流形中的曲面。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.