光滑函数的稳定器与轨道
Stabilizers and orbits of smooth functions
摘要：
设$f:R^m \to R$是一个光滑函数，使得$f(0)=0$。当对任意保向同构$ϕ:\mathbb{R} \to \mathbb{R}$使得$ϕ(0)=0$，函数$ϕ\circ f$右等价于$f$时，给出了$f$的一个条件，即存在一个同构$h:\mathbb{R}^m \to \mathbb{R}^m$使得$ϕ\circ f = f \circ h$在$0\in \mathbb{R}^m$ .要求$f$属于它的Jacobi理想。这个性质是相当普遍的：它对于奇点的稳定等价是不变的，并且对于非退化的临界点、简单奇点和许多其他点都成立。我们还将此结果全球化如下。设$M$是一个光滑紧致流形，$f:M \to [0,1]$是一个满射光滑函数，$\mathrm{Diff}(M)$是$M$的自同态群，而$\mathrm{Diff}^{[0,1]}(\mathbb{R})$是$\mathbb{R}$的自同态群，它们具有紧支集且保持$[0,1]$不变. $C^{\infty}(M,R)$上有两个自然的$\mathrm{Diff}(M)$和$\mathrm{Diff}(M) \times \mathrm{Diff}^{[0,1]}(\mathbb{R})$的右和左-右操作。令$S_M(f)$、$S_{MR}(f)$、$O_{M}(f)$和$O_{MR}(f)$是$f$相对于这些动作的相应稳定器和轨道。在对$f$的适当假设下，我们得到以下同伦等价$S_M(f) \approx S_{MR}(f)$和$O_M \approx O_{MR}$。对光滑映射$M \to S^1$也得到了类似的结果。
英文摘要：
Let $f:R^m \to R$ be a smooth function such that $f(0)=0$. We give a condition on $f$ when for arbitrary preserving orientation diffeomorphism $ϕ:\mathbb{R} \to \mathbb{R}$ such that $ϕ(0)=0$ the function $ϕ\circ f$ is right equivalent to $f$, i.e. there exists a diffeomorphism $h:\mathbb{R}^m \to \mathbb{R}^m$ such that $ϕ\circ f = f \circ h$ at $0\in \mathbb{R}^m$. The requirement is that $f$ belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated critical points, simple singularities and many others. We also globalize this result as follows. Let $M$ be a smooth compact manifold, $f:M \to [0,1]$ a surjective smooth function, $\mathrm{Diff}(M)$ the group of diffeomorphisms of $M$, and $\mathrm{Diff}^{[0,1]}(\mathbb{R})$ the group of diffeomorphisms of $\mathbb{R}$ that have compact support and leave $[0,1]$ invariant. There are two natural right and left-right actions of $\mathrm{Diff}(M)$ and $\mathrm{Diff}(M) \times \mathrm{Diff}^{[0,1]}(\mathbb{R})$ on $C^{\infty}(M,R)$. Let $S_M(f)$, $S_{MR}(f)$, $O_{M}(f)$, and $O_{MR}(f)$ be the corresponding stabilizers and orbits of $f$ with respect to these actions. Under mild assumptions on $f$ we get the following homotopy equivalences $S_M(f) \approx S_{MR}(f)$ and $O_M \approx O_{MR}$. Similar results are obtained for smooth mappings $M \to S^1$.
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