Special generic maps and Gromoll filtration

A smooth map of a closed $n$-dimensional manifold into $\mathbf{R}^p$ with $1 \leq p \leq n$ is a special generic map if it has only definite folds as its singularities. We show that for $1 \leq p < n$ and $n \geq 6$, 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 $p$.