Elimination of definite fold II

In this paper, we first give a new simple proof to the elimination theorem of definite fold by homotopy for generic smooth maps of manifolds of dimension strictly greater than $2$ into the $2$–sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but is also algorithmic: the procedures enable us to construct various explicit examples. We also study simple stable maps of $3$–manifolds into the $2$–sphere without definite fold. Furthermore, we prove the non-existence of singular Legendre fibrations on $3$–manifolds, answering negatively to a question posed in our previous paper.