  Special generic maps are higher dimensional versions of Morse functions with
exactly two singular points, characterizing spheres topologically except
$4$-dimensional cases: in these cases standard spheres are characterized.
Canonical projections of unit spheres are special generic. In suitable cases,
it is easy to construct special generic maps on manifolds represented as
connected sums of products of spheres for example. It is an interesting fact
that these maps restrict the topologies and the differentiable structures
admitting them strictly in various cases. For example, exotic spheres, which
are not diffeomorphic to standard spheres, admit no special generic map into
some Euclidean spaces in considerable cases.
  In general, it is difficult to find (families of) manifolds admitting no such
maps of suitable classes. The present paper concerns a new result on this work
where key objects are products of cohomology classes of the manifolds. We can
see that manifolds such as closed symplectic manifolds and real projective
spaces admit no special generic maps into any connected nin-closed manifold in
considerable cases.
