Let $M$ be a compact surface and $P$ be either $\mathbb{R}$ or $S^1$. For a smooth map $f:M\to P$ and a closed subset $V\subset M$, denote by $\mathcal{S}(f,V)$ the group of diffeomorphisms $h$ of $M$ preserving $f$, i.e. satisfying the relation $f\circ h = f$, and fixed on $V$. Let also $\mathcal{S}'(f,V)$ be its subgroup consisting of diffeomorphisms isotopic relatively $V$ to the identity map $\mathrm{id}_{M}$ via isotopies that are not necessarily $f$-preserving. The groups $\pi_0 \mathcal{S}(f,V)$ and $\pi_0 \mathcal{S}'(f,V)$ can be regarded as analogues of mapping class group for $f$-preserving diffeomorphisms. The paper describes precise algebraic structure of groups $\pi_0 \mathcal{S}'(f,V)$ and some of their subgroups and quotients for a large class of smooth maps $f:M\to P$ containing all Morse maps, where $M$ is orientable and distinct from $2$-sphere and $2$-torus. In particular, it is shown that for certain subsets $V$ "adapted" in some sense with $f$, the groups $\pi_0 \mathcal{S}'(f,V)$ are solvable and Bieberbach.

中文翻译：

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同质微分同胚在表面上的函数变形

令 $M$ 是一个紧凑曲面，而 $P$ 是 $\mathbb{R}$ 或 $S^1$。对于平滑映射 $f:M\to P$ 和闭子集 $V\subset M$，用 $\mathcal{S}(f,V)$ 表示 $M$ 的微分同胚群 $h$ 保留 $ f$，即满足关系$f\circ h = f$，并固定在$V$ 上。还设 $\mathcal{S}'(f,V)$ 是它的子群，由通过不一定是 $f$ 的同位素到恒等映射 $\mathrm{id}_{M}$ 的微分同胚组成-保存。群 $\pi_0 \mathcal{S}(f,V)$ 和 $\pi_0 \mathcal{S}'(f,V)$ 可以看作是保留 $f$ 的微分同胚的映射类群的类似物。该论文描述了群 $\pi_0 \mathcal{S}'(f,V)$ 及其一些子群和商的精确代数结构，用于包含所有莫尔斯图的一大类平滑映射 $f:M\to P$，其中 $M$ 是可定向的，与 $2$-sphere 和 $2$-torus 不同。特别地，它表明对于某些在某种意义上与 $f$“适应”的子集 $V$，群 $\pi_0\mathcal{S}'(f,V)$ 是可解的和 Bieberbach。