  As Reeb's theorem shows, Morse functions with exactly two singular points on
closed manifolds are very simple and important. They characterize spheres whose
dimensions are not $4$ topologically and the $4$-dimensional unit sphere.
  Special generic maps are generalized versions of these maps. Canonical
projections of unit spheres are special generic. Studies of Saeki and Sakuma
since the 1990s, followed by Nishioka and Wrazidlo, show that the
differentiable structures of the spheres and the homology groups of the
manifolds (in several classes) are restricted. We see special generic maps are
attractive.
  Our paper studies the cohomology rings of manifolds admitting such maps. As
our new result, we find a new family of manifolds whose cohomology rings are
similar and find that the (non-)existence of special generic maps are closely
related to the topologies. More explicitly, we have previously found related
families and our new manifolds add to these discoveries.
