Knotted Klein bottles with only double points (Q1411282)

The author considers a Klein bottle \(F\) embedded in \({\mathbb R}^4\) such that the singular set of the projection from \({\mathbb R}^4={\mathbb R}^3\times{\mathbb R}\) to \({\mathbb R}^3\times\{0\}\), restricted to \(F\), consists only of double points. The main result is that \(F\) is either a ribbon Klein bottle or is obtained from a spun Klein bottle by \(m\)-fusion. (These concepts are defined in the paper.) It is also shown that if every component of the singular set on \(F\) is not null-homotopic, and \(\pi_1\left({\mathbb R}^4\setminus F\right)\cong{\mathbb Z}_2\), then \(F\) bounds a solid Klein bottle in \({\mathbb R}^4\).