2024/12/24 by Arunima Ray, Ray, Arunima
Computer Science · Engineering · Mathematics · #57K40 #57N35 #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2412.18423
openalex publication_date 2024/12/24 · openalex created_date 2024/12/26 · openalex updated_date 2026/07/28
There are two main approaches to building locally flat embedded surfaces in 4-manifolds: direct methods which geometrically manipulate a given map of a surface, and more indirect methods using surgery theory. Both rely on Freedman-Quinn's disc embedding theorem. In this expository article, we give an introduction to these methods by sketching proofs of the following results: every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus (Lee-Wilczynski); and every Alexander polynomial one knot in S3 is topologically slice (Freedman-Quinn).