Nonexistence of homotopy equivalences which are 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.