Topologically trivial proper 2-knots

TL;DR

This paper explores exotic embeddings of noncompact surfaces in 4-manifolds, revealing uncountably many exotic planes with diverse properties and showing that all compact surfaces in the 4-ball have infinitely many smooth embeddings in R^4.

Contribution

It introduces two classes of exotic planes with distinct properties, constructs exotic planes with uncountable symmetry groups, and demonstrates the complexity of embeddings of compact surfaces in 4-manifolds.

Findings

Two uncountable classes of exotic planes with different properties

Every compact surface in the 4-ball has infinitely many smooth embeddings in R^4

Exotic planes can have uncountable symmetry groups injecting into the mapping class group

Abstract

We study smooth, proper embeddings of noncompact surfaces in 4-manifolds, focusing on exotic planes and annuli, i.e., embeddings pairwise homeomorphic to the standard embeddings of R^2 and R^2-int D^2 in R^4. We encounter two uncountable classes of exotic planes, with radically different properties. One class is simple enough that we exhibit explicit level diagrams of them without 2-handles. Diagrams from the other class seem intractable to draw, and require infinitely many 2-handles. We show that every compact surface embedded rel nonempty boundary in the 4-ball has interior pairwise homeomorphic to infinitely many smooth, proper embeddings in R^4. We also see that the almost-smooth, compact, embedded surfaces produced in 4-manifolds by Freedman theory must have singularities requiring infinitely many local minima in their radial functions. We construct exotic planes with uncountable group actions injecting into the pairwise mapping class group. This work raises many questions, some of which we list.