Homotopically equivalent smooth manifolds. I

In this paper we introduce a method for the investigation of smooth simply connected manifolds of dimension n ≥ 5 that permits a classification of them with exactness up to orientation-preserving diffeomorphisms. This method involves a detailed investigation of the properties of the so-called Thom complexes of normal bundles and is based on a theorem of Smale concerning the equivalence of the concepts of “h-cobordism” and “orientation-preserving diffeomorphism.” In the last chapter we work out some simple examples. Appendices are given in which the results of this paper are applied to certain other problems.