Equivariant Floer Homology, Concordance, and Homology Cobordism

Michigan State University, East Lansing MI

Investigators

Abstract

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.