  Closed (and simply-connected) manifolds whose dimensions are greater than 4
are classified via sophisticated algebraic and abstract theory such as surgery
theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed
manifolds in such ways. However, the latter work via geometric and constructive
ways is not so difficult. The assumption that the dimensions are not high
enables us to handle the manifolds via diagrams for example. It is difficult to
study higher dimensional manifolds in these ways, although it is natural and
important.
  In the present paper, we present such studies via {\it fold} maps, which are
higher dimensional variants of Morse functions. The author previously
constructed fold maps on 7-dimensional closed and simply-connected manifolds
satisfying additional conditions on cohomology rings, including so-called {\it
exotic} homotopy spheres. This paper concerns fold maps on such manifolds of a
wider class.
