Standard special generic maps of homotopy spheres into Euclidean spaces

By means of the technique of Stein factorization we introduce and study certain special generic maps of homotopy spheres into Euclidean spaces called standard. Modifying a construction due to Weiss, we show that standard special generic maps give naturally 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.