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

Abstract

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 $C^{\infty }$ 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 $n\ge 2$ even, there always exists a $C^{\infty }$ 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 $1\le k\le (n-5)/2$.

31 pages, 20 figures. Section 8 has been added. Some always-realizable moves have been modified