标题： 光滑函数将克莱因瓶分成两个莫比乌斯带 显示英文标题

标题： Smooth functions that split a Klein bottle into two Möbius bands

摘要： 给定一个紧致曲面$M$，考虑群$\mathcal{D}(M)$的$\mathcal{C}^{\infty}$微分同胚在空间$\mathcal{C}^{\infty}(M)$的$\mathcal{C}^{\infty}$函数上的右作用$\mathcal{C}^{\infty}(M)\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M)$，$(f, h) \mapsto f\circ h$，其中$M$是$M$上的函数。 对于$f\in\mathcal{C}^{\infty}(M)$用$\mathcal{O}(f)$表示其轨道，用$\mathcal{O}_f(f)$表示包含$f$的$\mathcal{O}(f)$的路径连通分支。 该论文继续了许多作者对紧致曲面上光滑函数的轨道$\mathcal{O}_f(f)$的同伦类型的计算系列。 我们在这里提供对克莱因瓶$K$上具有以下性质的函数$f\in\mathcal{C}^{\infty}(K)$的$\mathcal{O}_f(f)$计算：(i) 在每个临界点$f$处，它与某个齐次多项式光滑等价（例如） $f$是 Morse)，并且 (ii) 存在一个正则连通分支$\alpha$的水平集的$f$，使得$K\setminus\alpha$是两个开莫比乌斯带的不相交并集，其闭包为$M_1$和$M_2$。 设$f_i = f|_{M_i}$为$f$在莫比乌斯带$M_i$上的限制，$i=1,2$和$\mathcal{O}_{f_i}(f_i)$为$f_i$在上述$\mathcal{D}(M_i)$作用下的轨道中的路径连通分支。 $\mathcal{O}_{f_i}(f_i)$的可能同伦类型之前已经被明确计算出来。 我们证明$\mathcal{O}_f(f)$与$\mathcal{O}_{f_1}(f_1) \times \mathcal{O}_{f_2}(f_2)$同伦等价。 显示英文摘要

摘要： 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 $\alpha$ of a level set of $f$ such that $K\setminus\alpha$ is a disjoint union of two open M\"obius bands, with closures $M_1$ and $M_2$. Let $f_i = f|_{M_i}$ be the restriction of $f$ to the M\"{o}bius 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 $\mathcal{O}_{f_1}(f_1) \times \mathcal{O}_{f_2}(f_2)$.