  Fold maps are higher dimensional versions of Morse functions, which play
important roles in the studies of smooth manifolds, and such general maps also
have been fundamental tools in the studies of smooth manifolds by using generic
maps. In this paper, we study {\it simple} fold maps, which are fold maps such
that any connected component of the inverse image of each singular value
includes at most one singular point. More precisely, we consider simple fold
maps having simple structures locally or globally and show that the source
manifolds are bounded by (PL) manifolds obtained by considering the structures
of maps under appropriate conditions. Such studies are regarded as extensions
of results obtained by Saeki, Suzuoka etc. by 2005, which state that closed
manifolds admitting simple fold maps and more generally stable maps into
manifolds of lower dimensions without boundaries inverse images of whose
regular values are always disjoint unions of spheres are bounded by compact
manifolds obtained by observing the given maps.
