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

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