Bordism and regular homotopy of low-dimensional immersions
To obtain explicit generators for the groups of regular homotopy classes of immersions of the \(k\)-sphere in \(\mathbb{R}^ n\), the author defines an immersion invariant as an element of the \(k\)-th homotopy group of a Stiefel manifold. Using this invariant, he obtains generators for some groups of low-dimensional immersions. Among other things, he shows: Theorem. The standard (Froissart-Morin) eversion of \(S^ 2\) in \(\mathbb{R}^ 3\) has, as a track, an immersion of \(S^ 2\times [0,1]\) in \(\mathbb{R}^ 4\) whose ends are embedded \(S^ 2\text{s}\). Each of these bounds a 3-ball in \(\mathbb{R}^ 4\). Capping off the track with these 3-balls yields an immersion \(K: S^ 3\to \mathbb{R}^ 4\). Performing the eversion twice and capping off gives an immersion \(E: S^ 3\to \mathbb{R}^ 4\). The immersions \(E\) and \(K\) generate the group of regular homotopy classes of immersions of \(S^ 3\) in \(\mathbb{R}^ 4\).
generators for the groups of regular homotopy classes of immersions of the \(k\)-sphere in \(\mathbb{R}^ n\)
immersions of \(S^ 3\) in \(\mathbb{R}^ 4\)
scientific article; zbMATH DE number 168269