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.