Nonexistence of homotopy equivalences which are C∞ stable or of finite codimension

In this paper we will study the condition on dimension n under which there exist closed n-manifolds N, P (possibly N=P) and a homotopy equivalence N→P which is not homotopic to any C∞ stable map and to any map of finite codimension by combining the surgery theory and the nonexistence result for general maps which are C∞ stable or of finite codimension.