Special generic maps and Gromoll filtration

A smooth map of a closed n-dimensional manifold into Rp with 1≤p≤n is a special generic map if it has only definite folds as its singularities. We show that for 1≤p<n and n≥6, a homotopy n-sphere admits a special generic map into Rp with standard properties if and only if its Gromoll filtration is equal to p.