A new approach to light bulb tricks: Disks in 4-manifolds

2022/09/24 by Danica Kosanović, Peter Teichner, Kosanović, Danica +1
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2209.12015

Abstract

For a 4-manifold M and a knot k\colon\mathbbS¹\hookrightarrow∂ M with dual sphere G\colon\mathbbS²\hookrightarrow∂ M, we compute the set \mathbbD(M;k) of smooth isotopy classes of neat embeddings \mathbbD²\hookrightarrow M with boundary k, using an invariant going back to Dax. Moreover, we construct a group structure on \mathbbD(M;k) and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group \mathbbD(M;k) to the mapping class group of M.

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