Topologically trivial proper 2-knots

TL;DR

This work investigates smooth, proper embeddings of noncompact surfaces in $\mathbb{R}^4$, focusing on exotic planes and annuli and revealing two uncountable, structurally distinct classes. It develops and uses invariants $g^\infty$ and $\kappa^\infty$, along with Casson-handle and exotic $\mathbb{R}^4$ techniques, to construct and distinguish vast families of exotic planes, many of which are generated by unknots yet exhibit rich end-structure and symmetry. The paper demonstrates that every compact surface embedded rel nonempty boundary in the 4-ball has infinitely many exotic interiors in $\mathbb{R}^4$, and that Freedman’s almost-smooth surfaces typically require infinitely many local minima in their radial level diagrams. It links double branched covers, end sums, and group actions to produce uncountably many exotica and uncountable families parametrized by Cantor-type sets, while establishing intriguing constraints and questions about universality, equivalence relations, and the end behavior of these 4-manifold embeddings. Overall, the results illuminate deep interactions between Casson handles, exotic $\mathbb{R}^4$-theory, and smooth structure on noncompact 2-knots, with broad implications for the study of smooth proper embeddings in 4-manifolds.

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.

Figures (14)

• Figure 1: An exotic plane from Theorem \ref{['0g']}(a), as an infinite, recursive level diagram. The thick curves represent bunches of the indicated numbers of parallel strands.
• Figure 2: Whitehead doubles and kinky handles
• Figure 3: A kinky handle: The diagram represents the top boundary of a product with $I$. The core is the obvious immersed disk with three double points bounded by the pictured curve.
• Figure 4: Constructing a periodic end.
• Figure 5: Creating a singularity with a neighborhood system of Casson handles.

Theorems & Definitions (62)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Definition 2.2