Special generic maps and Gromoll filtration

TL;DR

This paper explores the relationship between special generic maps of homotopy spheres into Euclidean spaces and the Gromoll filtration, establishing a precise equivalence for spheres of dimension at least six.

Contribution

It characterizes when homotopy spheres admit special generic maps into Euclidean spaces based on their Gromoll filtration, linking differential topology and geometric mapping properties.

Findings

Homotopy spheres admit special generic maps iff their Gromoll filtration equals the target dimension.

The result holds for spheres of dimension at least six.

Provides a new criterion connecting Gromoll filtration to the existence of special generic maps.

Abstract

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 $p$.