Self-Referential Discs and the Light Bulb Lemma

TL;DR

This paper explores the complex behavior of self-referential discs in 4-manifolds, revealing new phenomena where homotopic and concordant discs are not isotopic, depending on the manifold's fundamental group.

Contribution

It introduces the concept of self-referential discs in 4-manifolds and demonstrates their non-isotopic nature in certain topological settings, expanding understanding of 4-manifold topology.

Findings

Existence of homotopic but not isotopic self-referential discs in certain 4-manifolds.

Discs with a common dual sphere can differ in isotopy class depending on the fundamental group.

Construction of a 3-ball in $S^2\times D^2\natural S^1\times B^3$ not isotopic to a standard embedding.

Abstract

We show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are homotopic rel $\partial$, concordant and coincide near their boundaries, yet are not properly isotopic. This occurs in manifolds without 2-torsion in their fundamental group, e.g. the boundary connect sum of $S^2\times D^2$ and $S^1\times B^3$, thereby exhibiting phenomena not seen with spheres. On the other hand we show that two such discs are isotopic rel $\partial$ if the manifold is simply connected. We construct in $S^2\times D^2\natural S^1\times B^3$ a properly embedded 3-ball properly homotopic to a $z_0\times B^3$ but not properly isotopic to $z_0\times B^3$.