A space level light bulb theorem in all dimensions

Abstract

Given a d -dimensional manifold M and a knotted sphere s\colon \mathbb{S}^{k-1}\hookrightarrow \partial M with 1\leq k\leq d , for which there exists a framed dual sphere G\colon \mathbb{S}^{d-k}\hookrightarrow \partial M , we show that the space of neat embeddings \mathbb{D}^{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.