Special generic maps and Gromoll filtration

Abstract

A smooth map of a closed n-dimensional manifold into R p with 1 ≤ p ≤ n is a special generic map if it has only definite folds as its singularities. We show that for 1 ≤ p < n and n ≥ 6, a homotopy n-sphere admits a special generic map into R p with standard properties if and only if its Gromoll filtration is equal to p.