Immersed surfaces and their lifts (Q2778272)

Let \(F\) be an orientable closed surface. Challenged by the fact that some immersed surfaces in \(\mathbb{R}^3\) are not projections of embedded surfaces in \(\mathbb{R}^4\) the author shows that for any immersion \(f:F\rightarrow \mathbb{R}^4\), it is possible to construct an immersion \(g:F \rightarrow \mathbb{R}^4\) which is the lift of \(f\) with respect to the projection of \(\mathbb{R}^4\) onto \(\mathbb{R}^3\), and furthermore, \(g\) is regularly homotopic to \(f\) (see the paper for the construction). In addition to that, for knotted surfaces in \(\mathbb{R}^4\) the so-called ``crossing number'' is introduced: If \(H\) is a regular homotopy from the knotted surface \(f\) to a trivially knotted surface \(g\) (in the sense of [\textit{F. Hosokawa} and \textit{A. Kawauchi}, Osaka J. Math. 16, 233-248 (1979; Zbl 0404.57020)]), then one considers the regular homotopy track \(\widehat{H}: I\times F \rightarrow I\times \mathbb{R}^4\) which is an immersion. The least number of components of the crossing set of \(\widehat {H}\) for all \(\widehat {H}\) is called crossing number, which turns out to be an isotopy invariant.