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Special-generic-like maps or SGL maps are introduced by the author motivatedby observing and investigating algebraic topological or differentialtopological properties of manifolds via nice smooth maps whose codimensions arenegative. The present paper says that manifolds admitting certain very explicitSGL maps are topologically restricted strongly and this also constructs thesemaps with the manifolds explicitly.  Morse functions with exactly two singular points on spheres or functions inReeb's theorem and canonical projections of naturally embedded spheres inEuclidean spaces are generali5zed as special generic maps. Their nice globalstructures motivate us to study such maps and the manifolds. Manifoldsrepresented as connected sums of the product of spheres and similar ones admitsuch maps in considerable cases. The topologies and the differentiablestructures of the manifolds of the domains of such maps are also stronglyrestricted: due to Saeki and Sakuma, followed by Nishioka, Wrazidlo and theauthor. Respecting their nice global structures, these maps are generalized andthis yields SGL maps.

Restrictions on manifolds admitting certain explicit special-generic-like maps and construction of maps with the manifolds