Fibers of generic maps on surfaces

Using special triangulations of a compact 2-dimensional topological manifold without boundary S, for every closed subset F⊆S we construct a dense in the mapping space C(F,[0,1]) family of piecewise linear mappings whose fibers consist of components homeomorphic to subcontinua of the figure eight. The number of fibers with a figure-eight component is evaluated for each such map in the case F=S. We then prove that every fiber of a generic map in C(F,[0,1]) consists only of components being either a singleton or a figure-eight-like hereditarily indecomposable continuum. This extends a result of Z. Buczolich and U.B. Darji.