Spaces of immersion of elementary fiber bundles (Q1406454)

Let \(\pi:{\mathcal E}\to{\mathcal B}\) be some mapping which places into correspondence to the immersion of the disk \(D^k\) the immersion of its boundary \(S^{k-1}\). The author cites without proof the analogue of the Smale theorem on the covering homotopy (see \textit{S. Smale} [Ann. Math. (2) 69, 2, 327--344 (1959; Zbl 0089.18201)]) for the case of fibered immersions. The mapping \(\widehat\pi:{\mathcal E}\to{\mathcal C}\) (\(m\geq k\), \(n> l\)) and the composition \(\widetilde\pi\circ \widetilde\pi:{\mathcal E}\to{\mathcal D}\) (\(m> k\), \(n> l\)) are the fiberings in the Serre sense.