Smooth functions that split a Klein bottle into two Möbius bands
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2025
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| author | Mazhar, Bohdan Maksymenko, Sergiy |
| author_facet | Mazhar, Bohdan Maksymenko, Sergiy |
| contents | Given a compact surface $M$, consider the right action $\mathcal{C}^{\infty}(M)\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M)$, $(f, h) \mapsto f\circ h$, of the group $\mathcal{D}(M)$ of $\mathcal{C}^{\infty}$ diffeomorphisms of $M$ on the space $\mathcal{C}^{\infty}(M)$ of $\mathcal{C}^{\infty}$ functions on $M$. For $f\in\mathcal{C}^{\infty}(M)$ denote by $\mathcal{O}(f)$ its orbit, and by $\mathcal{O}_f(f)$ the path component of $\mathcal{O}(f)$ containing $f$.
The paper continues a series of computations by many authors of homotopy types of orbits $\mathcal{O}_f(f)$ of smooth functions on compact surfaces. We provide here the computations of $\mathcal{O}_f(f)$ for a special class of functions $f\in\mathcal{C}^{\infty}(K)$ on the Klein bottle $K$ having the following properties: (i) at each critical point $f$ is smoothly equivalent to some homogeneous polynomial (e.g. $f$ is Morse), and (ii) there is a regular connected component $α$ of a level set of $f$ such that $K\setminusα$ is a disjoint union of two open Möbius bands, with closures $M_1$ and $M_2$. Let $f_i = f|_{M_i}$ be the restriction of $f$ to the Möbius band $M_i$, $i=1,2$, and $\mathcal{O}_{f_i}(f_i)$ be the path component of $f_i$ in its orbit with respect to the above action of $\mathcal{D}(M_i)$. The possible homotopy types of $\mathcal{O}_{f_i}(f_i)$ are explicitly computed earlier. We prove that $\mathcal{O}_f(f)$ is homotopy equivalent to $\mathcal{O}_{f_1}(f_1) \times \mathcal{O}_{f_2}(f_2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_19636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Smooth functions that split a Klein bottle into two Möbius bands Mazhar, Bohdan Maksymenko, Sergiy Geometric Topology Algebraic Topology Differential Geometry 57S05, 57R45, 37C05 Given a compact surface $M$, consider the right action $\mathcal{C}^{\infty}(M)\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M)$, $(f, h) \mapsto f\circ h$, of the group $\mathcal{D}(M)$ of $\mathcal{C}^{\infty}$ diffeomorphisms of $M$ on the space $\mathcal{C}^{\infty}(M)$ of $\mathcal{C}^{\infty}$ functions on $M$. For $f\in\mathcal{C}^{\infty}(M)$ denote by $\mathcal{O}(f)$ its orbit, and by $\mathcal{O}_f(f)$ the path component of $\mathcal{O}(f)$ containing $f$. The paper continues a series of computations by many authors of homotopy types of orbits $\mathcal{O}_f(f)$ of smooth functions on compact surfaces. We provide here the computations of $\mathcal{O}_f(f)$ for a special class of functions $f\in\mathcal{C}^{\infty}(K)$ on the Klein bottle $K$ having the following properties: (i) at each critical point $f$ is smoothly equivalent to some homogeneous polynomial (e.g. $f$ is Morse), and (ii) there is a regular connected component $α$ of a level set of $f$ such that $K\setminusα$ is a disjoint union of two open Möbius bands, with closures $M_1$ and $M_2$. Let $f_i = f|_{M_i}$ be the restriction of $f$ to the Möbius band $M_i$, $i=1,2$, and $\mathcal{O}_{f_i}(f_i)$ be the path component of $f_i$ in its orbit with respect to the above action of $\mathcal{D}(M_i)$. The possible homotopy types of $\mathcal{O}_{f_i}(f_i)$ are explicitly computed earlier. We prove that $\mathcal{O}_f(f)$ is homotopy equivalent to $\mathcal{O}_{f_1}(f_1) \times \mathcal{O}_{f_2}(f_2)$. |
| title | Smooth functions that split a Klein bottle into two Möbius bands |
| topic | Geometric Topology Algebraic Topology Differential Geometry 57S05, 57R45, 37C05 |
| url | https://arxiv.org/abs/2508.19636 |