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Fold maps are higher dimensional versions of Morse functions, which playimportant roles in the studies of smooth manifolds, and such general maps alsohave been fundamental tools in the studies of smooth manifolds by using genericmaps. In this paper, we study {\it simple} fold maps, which are fold maps suchthat any connected component of the inverse image of each singular valueincludes at most one singular point. More precisely, we consider simple foldmaps having simple structures locally or globally and show that the sourcemanifolds are bounded by (PL) manifolds obtained by considering the structuresof maps under appropriate conditions. Such studies are regarded as extensionsof results obtained by Saeki, Suzuoka etc. by 2005, which state that closedmanifolds admitting simple fold maps and more generally stable maps intomanifolds of lower dimensions without boundaries inverse images of whoseregular values are always disjoint unions of spheres are bounded by compactmanifolds obtained by observing the given maps.

Simple fold maps and manifolds bounded by the source manifolds