تقرير
Simplifying generic smooth maps to the 2-sphere and to the plane
| العنوان: | Simplifying generic smooth maps to the 2-sphere and to the plane |
|---|---|
| المؤلفون: | Saeki, Osamu |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Geometric Topology, 57R45 (Primary) 58K30 (Secondary) |
| الوصف: | 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 $1 \leq k < (n-3)/2$. Comment: 16 pages, 19 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.10145 |
| رقم الأكسشن: | edsarx.2407.10145 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |