Standard Special Generic Maps of Homotopy Spheres into Euclidean Spaces

Dominik Wrazidlo

Published 2017-07-26Version 1

A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki to determine the set of integers $p$ for which a given homotopy sphere admits a special generic map into $\mathbb{R}^{p}$. 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. As an application of our result we deduce the (non-)existence of standard special generic maps on some concrete homotopy spheres such as Milnor spheres.