In this talk I will present recent progress in the following problem raised by O. Saeki in 1993. Determine the set of integers \(p\) for which a given homotopy sphere admits a special generic map into \(\mathbb{R}^p\). Here, a so-called special generic map is by definition a map between smooth manifolds all of whose singularities are definite fold points.

By means of the technique of Stein factorization we introduce and study certain standard special generic maps of homotopy spheres into Euclidean spaces. Modifying a construction due to Weiss, we show that standard special generic maps naturally give rise to a filtration of the group of homotopy spheres by subgroups that is strongly related to the Gromoll filtration. Finally, we apply our result to some concrete homotopy spheres, which in particular answers Saeki's problem for the Milnor 7-sphere.