Self-referential discs and the light bulb lemma

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 properly homotopic rel $\partial$ and coincide near their boundaries, yet are not properly isotopic. This occurs in manifolds without 2-torsion in their fundamental group, thereby exhibiting phenomena not seen with spheres, e.g. the boundary connect sum of $S^2\times D^2$ and $S^1\times B^3$. 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 $z_0\times B^3$.