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