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We show that manifolds admitting special generic maps also admit nicegeneralized multisections.  Special generic maps are natural generalized versions of Morse functions withexactly two singular points on closed manifolds, characterizing spheres whosedimensions are not $4$ topologically and the $4$-dimensional unit sphere, andcanonical projections of unit spheres. They are shown to restrict thedifferentiable structures of spheres etc. and topologies of more generalmanifolds strongly by Saeki, Sakuma etc., followed by Nishioka, Wrazidlo etc.and followed by the author. Some elementary or important manifolds also admitsuch maps.  (Generalized) multisections of manifolds are nice decompositions of (compact)manifolds, generalizing so-called Heegaard splittings of $3$-dimensionalmanifolds.  PL manifolds have been shown to have (generalized) multisections enjoyingcertain properties by Rubinstein and Tillmann.

Manifolds admitting special generic maps and their nice generalized multisections