“…The notion of singular fibers of C ∞ maps between manifolds without boundary was first introduced in [7], where classifications of singular fibers of stable maps M → N with (dim M, dim N ) = (2, 1), (3,2) and (4,3) were obtained. Later, singular fibers of stable maps of manifolds without boundary were studied in [7,8,11,12,16,17,18], especially in connection with cobordisms. The first author [7] established the theory of universal complex of singular fibers of C ∞ maps as an analogy of the Vassiliev complex for map germs [6,15].…”