Simplifying generic smooth maps to the 2-sphere and to the plane

We study how to construct explicit deformations of generic smooth maps from closed n--dimensional manifolds M with n ≥ 4 to the $2$--sphere S² and show that every smooth map M → S² is homotopic to a C^∞ stable map with at most one cusp point and with only folds of the middle absolute index. Furthermore, if n is even, such a C^∞ stable map can be so constructed that the restriction to the singular point set is a topological embedding. As a corollary, we show that for n ≥ 4 even, there always exists a C^∞ stable map M → R² with at most one cusp point such that the restriction to the singular point set is a topological embedding. As another corollary, we give a new proof to the existence of an open book structure on odd dimensional manifolds which extends a given one on the boundary, originally due to Quinn. Finally, using the open book structure thus constructed, we show that k--connected n--dimensional manifolds always admit a fold map into R² without folds of absolute indices i with 1 ≤ i ≤ k, for n ≥ 7 odd and 1 ≤ k < (n-3)/2.