Smooth maps compatible with simplicial structures and inverse images

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.