  The class of special generic maps is a natural class of smooth maps
containing Morse functions on spheres with exactly two singular points and
canonical projections of unit spheres. We find new restrictions on such maps on
$6$-dimensional or higher dimensional closed and simply-connected manifolds
into ${\mathbb{R}}^5$.
  Spheres which are not diffeomorphic to unit spheres do not admit such maps
whose codimensions are negative in considerable cases. They restrict the
homeomorphism and the diffeomorphism types of the manifolds in general. On the
other hands, some elementary manifolds admit special generic maps into suitable
Euclidean spaces: manifolds represented as connected sums of products of unit
spheres are of such examples. This motivates us to study the (non-)existence of
special generic maps on elementary manifolds such as projective spaces and some
closed and simply-connected manifolds. For example, new explicit investigations
of cohomology rings are keys in our new study.
