Deformations of functions on surfaces by isotopic to the identity diffeomorphisms

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.

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