Maps with only Morin singularities and the Hopf invariant one problem

We show that the non-existence of elements in the p-stem π_p^S of Hopf invariant one implies that: there exists no smooth map f: M → N with only fold singularities when M is a closed n-dimensional manifold with odd Euler characteristic and N is an almost parallelizable p-dimensional manifold (n ≥ p), provided that p ≠ 1, 3, 7. In fact, the result itself is originally due to Kikuchi and Saeki [25, 34]. Our proof clarifies the relationship between the two problems and gives a new insight to the problem of the global singularity theory. Furthermore we generalize the above result to maps with only Morin singularities of types A_k with k ≤ 3 when p ≠ 1, 2, 3, 4, 7, 8.