Subvarieties of codimensionality one and of simple homotopic type

Let $f_1: M_1\to M_2$ be a homotopic equivalence and $N\subset M_2$ a subvariety of codimensionality one. In this paper we indicate the conditions under which there is a mapping $f_2\sim f_1$ such that the variety $f_2^{-1}(N)$ is simply homotopically equivalent to the variety $N$.