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

TL;DR

This paper introduces a novel method for classifying disk embeddings in 4-manifolds with boundary knots, utilizing an invariant from Dax and revealing a complex, often non-abelian group structure.

Contribution

It computes the set of isotopy classes of disks with boundary knots in 4-manifolds, constructs a group structure on this set, and relates it to the mapping class group, extending previous sphere results.

Findings

The set of disk isotopy classes is computed using a Dax-based invariant.

A group structure on disk classes is established, often non-abelian and infinitely generated.

Connections to the mapping class group of the 4-manifold are demonstrated.

Abstract

For a 4-manifold $M$ and a knot $k\colon\mathbb{S}^1\hookrightarrow\partial M$ with dual sphere $G\colon\mathbb{S}^2\hookrightarrow\partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings $\mathbb{D}^2\hookrightarrow M$ with boundary $k$, using an invariant going back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ 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 $\mathbb{D}(M;k)$ to the mapping class group of $M$.