Decompositions of manifolds into submanifolds compatible with specific fold maps

We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the differentibale structures of manifolds globally, decomposing manifolds are important and this presents interesting topics and problems on geometry of manifolds. The notion of a Heegaard splitting of a $3$-dimensional closed and connected manifold presents a pioneering study. A $3$-dimensional closed and connected manifold is always decomposed into two copies of a so-called $3$-dimensional handlebody via a so-called Heegaard surface, which is a closed and connected surface. Heegaard splitiings are generalized as multisections of smooth or PL manifolds in the 2010s. As a way of understanding, these decompositions are understood via Morse functions and general generic smooth maps whose codimensions are negative.