A homotopy invariant of stable maps to oriented surfaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915408616882176 |
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| author | Kahmeyer, Liam Sadykov, Rustam |
| author_facet | Kahmeyer, Liam Sadykov, Rustam |
| contents | The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $Σ(f)$. We study the parity of the number of components of $Σ(f)$. The image $f(Σ)$ of the singular set inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to the cumulative winding number $ω(f)\in \frac{1}{2}\mathbb{Z}$ of $Σ(f)$. When the dimension of the manifold $M$ is even we also define an invariant $I(f)$ which is the residue class modulo $4$ of the sum of the number of components of $Σ(f)$, the number of cusps, and twice the number of self-intersection points of $f(Σ)$. Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $Σ(f)$ does not change under homotopy of $f$ provided that one of the following conditions is satisfied: (i) the dimension of $M$ is even, (ii) the singular set of the homotopy is an orientable manifold, or (iii) the image of the singular set of the homotopy does not have triple self-intersection points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_07297 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A homotopy invariant of stable maps to oriented surfaces Kahmeyer, Liam Sadykov, Rustam Geometric Topology 58K30, 58K65, 57R45 The singular set of a generic map $f: M\to F$ of a manifold $M$ of dimension $m\ge 2$ to an oriented surface $F$ is a closed smooth curve $Σ(f)$. We study the parity of the number of components of $Σ(f)$. The image $f(Σ)$ of the singular set inherits canonical local orientations via so-called chessboard functions. Such a local orientation gives rise to the cumulative winding number $ω(f)\in \frac{1}{2}\mathbb{Z}$ of $Σ(f)$. When the dimension of the manifold $M$ is even we also define an invariant $I(f)$ which is the residue class modulo $4$ of the sum of the number of components of $Σ(f)$, the number of cusps, and twice the number of self-intersection points of $f(Σ)$. Using the cumulative winding number and the invariant $I(f)$, we show that the parity of the number of connected components of $Σ(f)$ does not change under homotopy of $f$ provided that one of the following conditions is satisfied: (i) the dimension of $M$ is even, (ii) the singular set of the homotopy is an orientable manifold, or (iii) the image of the singular set of the homotopy does not have triple self-intersection points. |
| title | A homotopy invariant of stable maps to oriented surfaces |
| topic | Geometric Topology 58K30, 58K65, 57R45 |
| url | https://arxiv.org/abs/2208.07297 |