  We study how to construct explicit deformations of generic smooth maps from
closed $n$--dimensional manifolds $M$ with $n \geq 4$ to the $2$--sphere $S^2$
and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable
map with at most one cusp point and with only folds of the middle absolute
index. Furthermore, if $n$ is even, such a $C^\infty$ 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 $n \geq 4$ even, there always
exists a $C^\infty$ stable map $M \to \mathbb{R}^2$ 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 $\mathbb{R}^2$ without folds of absolute indices $i$ with
$1 \leq i \leq k$, for $n \geq 7$ odd and $1 \leq k < (n-3)/2$.
