  As a higher dimensional version of the theory of Morse functions, there have
been various studies of smooth manifolds using generic smooth maps. As
fundamental results, in these studies, they have found that inverse images of
such maps often restrict the types of the source manifolds. For example, if a
generic map such that the inverse image of a regular value is not
null-cobordant, then the homology group of its {\it Reeb} space, which is
defined as the space of all the connected components of inverse images of the
map and a fundamental tool in the theory of generic smooth maps, is known to be
non-trivial. In this paper, we show similar results in new appropriate
situations. These works are regarded as extensions of works by Hiratuka and
Saeki in 2013--4.
