  Constructing Morse functions and their higher dimensional versions or fold
maps is fundamental, important and challenging in investigating the topologies
and the differentiable structures of differentiable manifolds via Morse
functions, fold maps and more general generic maps. It is one of important and
interesting branches of the singularity theory of differentiable maps and
applications to geometry of manifolds. In this paper we present fold maps with
information of cohomology rings of their Reeb spaces. Reeb spaces are defined
as the spaces of all connected components of all preimages, and in suitable
situations inherit topological information such as homology groups and
cohomology rings of the manifolds. Previously, the author demonstrated
construction of fold maps in various cases : key methods are surgery operations
to manifolds and maps and in this paper, we present more useful surgery
operations and by them we construct new fold maps. More precisely, fold maps
with singular value sets with crossings: the singular value set of a smooth map
is the image of the set of all singular points and note that for fold maps, the
set of all singular points are closed submanifolds without boundaries and the
restrictions to them are immersions of codimension 1.
