  In our paper, we introduce special-generic-like maps or SGL maps as smooth
maps and study their several algebraic topological and differential topological
properties.
  The new class generalize the class of so-called special generic maps. Special
generic maps are smooth maps which are locally projections or the product maps
of Morse functions and the identity maps on disks. Morse functions with exactly
two singular points on spheres or Morse functions in Reeb's theorem are
simplest examples. Special generic maps and the manifolds of their domains have
been studied well. Their structures are simple and this help us to study
explicitly. As important properties, they have been shown to restrict the
topologies and the differentiable structures of the manifolds strongly by Saeki
and Sakuma, followed by Nishioka, Wrazidlo and the author. To cover wider
classes of manifolds as the domains, the author previously introduced a class
generalizing the class of special generic maps and smaller than our class:
simply generalized special generic maps.
