  Special-generic-like maps or SGL maps are introduced by the author motivated
by observing and investigating algebraic topological or differential
topological properties of manifolds via nice smooth maps whose codimensions are
negative. The present paper says that manifolds admitting certain very explicit
SGL maps are topologically restricted strongly and this also constructs these
maps with the manifolds explicitly.
  Morse functions with exactly two singular points on spheres or functions in
Reeb's theorem and canonical projections of naturally embedded spheres in
Euclidean spaces are generali5zed as special generic maps. Their nice global
structures motivate us to study such maps and the manifolds. Manifolds
represented as connected sums of the product of spheres and similar ones admit
such maps in considerable cases. The topologies and the differentiable
structures of the manifolds of the domains of such maps are also strongly
restricted: due to Saeki and Sakuma, followed by Nishioka, Wrazidlo and the
author. Respecting their nice global structures, these maps are generalized and
this yields SGL maps.
