Exotic families of embeddings
Dave Auckly, Daniel Ruberman

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 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.
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Taxonomy
TopicsAdvanced Topics in Algebra · advanced mathematical theories · Cellular Automata and Applications
