Exotic families of embeddings

TL;DR

This paper constructs various topologically trivial yet smoothly non-trivial families of 3-manifold embeddings in 4-manifolds, revealing new phenomena in smooth topology and embedding theory.

Contribution

It introduces novel methods to produce smoothly non-trivial families of embeddings that are topologically trivial, including embeddings of homology spheres in S^4 and families in blown-up K3 surfaces.

Findings

Existence of non-isotopic embeddings with diffeomorphic complements

Construction of high-dimensional families of embeddings

A technique to convert non-trivial embedding families into non-trivial submanifold families

Abstract

We construct a number of topologically trivial but smoothly non-trivial families of embeddings of 3-manifolds in 4-manifolds. These include embeddings of homology spheres in $S^4$ that are not isotopic but have diffeomorphic complements, and families (parameterized by high-dimensional spheres) of embeddings of any 3-manifold that embeds in a blown-up K3 surface. In each case, the families are constructed so as to be topologically trivial in an appropriate sense. We also illustrate a general technique for converting a non-trivial family of embeddings into a non-trivial family of submanifolds.