4-dimensional analogues of Dehn's lemma

TL;DR

This paper explores 4-dimensional analogues of Dehn's lemma, showing conditions under which spheres and tori in 4-manifolds bound embedded balls or tori, highlighting differences between smooth and topological categories.

Contribution

It provides new examples and results on when spheres and tori in 4-manifolds bound embedded or immersed balls and tori, revealing distinctions between smooth and topological embeddings.

Findings

Essential 2-spheres may not extend to embedded balls in smooth 4-manifolds.

In topological category, certain spheres bound embedded balls in homology sphere boundaries.

Constructs examples of incompressible tori that extend to maps but not embeddings in 4-manifolds.

Abstract

We investigate certain $4$-dimensional analogues of the classical $3$-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential $2$-sphere $S$ in the boundary of a simply connected $4$-manifold $W$ such that $S$ is null-homotopic in $W$ need not extend to an embedding of a ball in $W$. However, if $W$ is simply connected (or more generally a $4$-manifold with abelian fundamental group) with boundary a homology sphere, then $S$ bounds a topologically embedded ball in $W$. Moreover, we give examples where such an $S$ does not bound any smoothly embedded ball in $W$. In a similar vein, we construct incompressible tori $T\subseteq \partial W$ where $W$ is a contractible $4$-manifold such that $T$ extends to a map of a solid torus in $W$, but not to any embedding of a solid torus in $W$. Moreover, we construct an incompressible torus $T$ in the boundary of a contractible $4$-manifold $W$ such that $T$ extends to a topological embedding of a solid torus in $W$ but no smooth embedding. As an application of our results about tori, we address a question posed by Gompf about extending certain families of diffeomorphisms of $3$-manifolds which he has recently used to construct infinite corks.