Four-dimensional analogues of Dehn's lemma

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 has abelian fundamental group with boundary a homology sphere, then S bounds a topologically embedded ball in W. Additionally, 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⊆∂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; he has recently used such families to construct infinite order corks.