Standard Special Generic Maps of Homotopy Spheres into Euclidean Spaces

TL;DR

This paper introduces standard special generic maps of homotopy spheres into Euclidean spaces, linking them to the Gromoll filtration and solving Saeki's problem for the Milnor 7-sphere.

Contribution

It defines and studies standard special generic maps, establishing their connection to the Gromoll filtration and addressing Saeki's open problem for specific homotopy spheres.

Findings

Standard special generic maps induce a filtration of the homotopy sphere group.

The approach relates to the Gromoll filtration and provides new insights into homotopy sphere mappings.

Saeki's problem is solved for the Milnor 7-sphere.

Abstract

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 in 1993 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. Finally, we apply our result to concrete homotopy spheres, which particularly answers Saeki's problem for the Milnor $7$-sphere.