Simplifying generic smooth maps to the 2-sphere and to the plane
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918037425225728 |
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| author | Saeki, Osamu |
| author_facet | Saeki, Osamu |
| contents | We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ 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 $n \geq 2$ 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 $1 \leq i \leq k$, for $n \geq 7$ odd and $1 \leq k \leq (n-5)/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10145 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simplifying generic smooth maps to the 2-sphere and to the plane Saeki, Osamu Geometric Topology 57R45 (Primary) 58K30, 57R70, 58K05 (Secondary) We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ 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 $n \geq 2$ 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 $1 \leq i \leq k$, for $n \geq 7$ odd and $1 \leq k \leq (n-5)/2$. |
| title | Simplifying generic smooth maps to the 2-sphere and to the plane |
| topic | Geometric Topology 57R45 (Primary) 58K30, 57R70, 58K05 (Secondary) |
| url | https://arxiv.org/abs/2407.10145 |