  Morse functions with exactly two singular points on spheres and canonical
projections of spheres belong to the class of a certain good class of smooth
maps: special generic maps. We mainly investigate information on cohomology of
closed and simply-connected manifolds admitting such maps into the
$5$-dimensional Euclidean spaces by investigating the embedded curves and
submanifolds and their preimages.
  Studies on homology groups for ones into the Euclidean spaces (whose
dimensions are lower than $5$ in most cases) have been pioneered by Saeki and
Sakuma since 1990s and later by Nishioka and Wrazidlo since 2010s. Recently the
author has started pioneering studies on the cohomology for cases where the
dimensions of the Euclidean spaces may not be lower than $5$. Our new cases are
difficult due to the situation that the dimensions of manifolds we consider are
higher. Previously, we have found several restrictions on the cohomology rings.
We present new restrictions by new investigations.
