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 $n\ge 2$ to the $2$–sphere $S^2$ and show that every smooth map $M\to S^2$ is homotopic to a $C^{\mathrm{\infty }}$ stable map with at most one cusp point and with only folds of the middle absolute index. Furthermore, if $n$ is even, such a $C^{\mathrm{\infty }}$ 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 $n\ge 2$ even, there always exists a $C^{\mathrm{\infty }}$ stable map $M\to {\mathbf{R}}^2$ 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 ${\mathbf{R}}^2$ without folds of absolute indices $i$ with $1\le i\le k$, for $n\ge 7$ odd and $1\le k\le (n-5)\mathrm{/}2$.