A space level light bulb theorem in all dimensions

TL;DR

This paper proves a generalized light bulb theorem for all dimensions, showing how embedding spaces of disks in manifolds simplify via handle attachments, with applications to low-dimensional topology and homotopy groups.

Contribution

It introduces a new delooping technique for embedding spaces in all dimensions using handle attachments, extending classical light bulb theorems.

Findings

Homotopy type of embedding spaces is simplified by increasing codimension.

Describes the first nontrivial homotopy group of these spaces in degree d-2k.

Provides tools for classifying disks in 4-manifolds with specific boundary conditions.

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.