Special generic maps and Gromoll filtration

A smooth map of a closed $n$-dimensional manifold into ${\mathbf{R}}^p$ with $1\le p\le n$ is a special generic map if it has only definite folds as its singularities. We show that for $1\le p<n$ and $n\ge 6$, a homotopy $n$-sphere admits a special generic map into ${\mathbf{R}}^p$ with standard properties if and only if its Gromoll filtration is equal to $p$.