On Eversion of Spheres

The celebrated Smale–Hirsch classification of immersions allows one to obtain several nice applications of algebraic topology to differential topology. Unfortunately, these applications have yet to be presented in books or survey papers either in Russian or in English. The purpose of this paper is to expose the simplest and most fundamental of these applications: the Smale–Kaiser theorem on the dimension of spheres that can be turned inside out, the Haefliger–Hirsch classification of immersions by means of equivariant maps, and its corollary concerning embeddings of highly connected manifolds (in particular, of spheres).