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.