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Special generic maps are higher dimensional versions of Morse functions withexactly 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 asconnected sums of products of spheres for example. It is an interesting factthat these maps restrict the topologies and the differentiable structuresadmitting them strictly in various cases. For example, exotic spheres, whichare not diffeomorphic to standard spheres, admit no special generic map intosome Euclidean spaces in considerable cases.  In general, it is difficult to find (families of) manifolds admitting no suchmaps of suitable classes. The present paper concerns a new result on this workwhere key objects are products of cohomology classes of the manifolds. We cansee that manifolds such as closed symplectic manifolds and real projectivespaces admit no special generic maps into any connected nin-closed manifold inconsiderable cases.

Closed manifolds admitting no special generic maps whose codimensions are negative and their cohomology rings