A space level light bulb theorem in all dimensions

2021/05/27 by Danica Kosanović, Peter Teichner, Kosanović, Danica +1 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2105.13032

Abstract

Given a d-dimensional manifold M and a knotted sphere s\colon\mathbbS^k-1\hookrightarrow∂ M with 1≤ k≤ d, for which there exists a framed dual sphere G\colon\mathbbS^d-k\hookrightarrow∂ M, we show that the space of neat embeddings \mathbbD^k\hookrightarrow M with boundary s can be delooped by the space of neatly embedded (k-1)-disks, with a normal vector field, in the d-manifold obtained from M by attaching a handle to G. This increase in codimension significantly simplifies the homotopy type of such embedding spaces, and is of interest also in low-dimensional topology. In particular, we apply the work of Dax to describe the first interesting homotopy group of these embedding spaces, in degree d-2k. In a separate paper we use this to give a complete isotopy classification of 2-disks in a 4-manifold with such a boundary dual.

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• On the detection of knotted spheres by their traces in high dimensions

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