Standard special generic maps of homotopy spheres into Euclidean spaces

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 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.

The 3-dimensional complex projective space admits no special generic maps

“…This is based on the theory on 7-dimensional homotopy spheres and special generic maps on them, referred in the first section, before the appearance of [15]. After [15] appeared, [36] announced another proof. It investigates restrictions on the torsion group of the integral homology group for a closed and connected smooth manifold whose rational homology group is isomorphic to that of a sphere.…”

“…However we omit this considering the main content of the present paper. We only introduce [2,22,23,34] and preprints [13,15,16,17] by the author, which review related theory. In the third section, we show a new result as Theorem 3 and have Main Theorem as Corollary 2.…”

Restrictions on special generic maps into ${\mathbb{R}}^5$ on $6$-dimensional or higher dimensional closed and simply-connected manifolds

“…According to a slide [32] of a related talk in a conference, Corollary 1 had been first shown for the 7-dimensional real projective space by using known facts and theory on 7-dimensional homotopy spheres and special generic maps in Theorem 1 before [15] appeared. After [15] was announced, [33] announced a proof investigating restrictions on the torsion groups of the integral homology groups for rational homology spheres: a rational homology sphere M is a closed and smooth manifold whose rational homology group H * (M ; Q) is isomorphic to that of a sphere.…”

“…Theorem 3 also implies that the existence of a special generic map f : M → R n on a 6-dimensional closed and simply-connected manifold M into the n-dimensional Euclidean space R n yields the vanishing of the cup product c 1 ∪ c 2 of any pair of two cohomology classes c 1 , c 2 ∈ H 2 (M ; Z) for n = 1, 2, 3, 4 and this implies that Main theorem 3 is a version for n = 5 of this. Special generic maps on closed and simply-connected manifolds were studied first in [22], followed by [21] and [31] for example (Theorems 1 and 2). These studies are essentially on such maps on spheres whose dimensions are arbitrary and closed and simplyconnected manifolds whose dimensions are at most 5.…”

The topologies and the differentiable structures of the images of special generic maps having simple structures

“…Example 1 ( [2], [32], [33], [34], [40], and so on.). An m-dimensional homotopy sphere always admits a special generic map into R 2 for m = 1, 4.…”