Spaces of immersions of elementary fibre bundles

S. M. Tuleshev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 1, pp. 173-197

Résumé

In the paper we consider immersions of elementary fibre objects. For fibre disk, i. e. a trivial fibre bundle of type $D^k\times D^l\to D^k$, we give a definition of fibrewise immersion and for fibre boundary spheres (fibre bundles $\partial D^k\times D^l\to\partial D^k$, $D^k\times\partial D^l\to D^k$ and $\partial D^k\times\partial D^l\to\partial D^k$) we define the notion of framed fibrewise immersions. It is proved that the natural maps from the space of immersions of fibre disk into spaces of framed immersions of fibre boundary spheres satisfy the axiom of the covering homotopy, i. e. are Serre bundles. This result is an initial step in solving the problem on the homotopy description of space of immersions of one fibre manifold into another fibre manifold.