Deformations of functions on surfaces by isotopic to the identity diffeomorphisms

Let M be a compact surface and P be either R or S1. For a smooth map f:M→P and a closed subset V⊂M, denote by S(f,V) the group of diffeomorphisms h of M preserving f, i.e. satisfying the relation f∘h=f, and fixed on V. Let also S′(f,V) be its subgroup consisting of diffeomorphisms isotopic relatively V to the identity map idM via isotopies that are not necessarily f-preserving. The groups π0S(f,V) and π0S′(f,V) can be regarded as analogues of mapping class group for f-preserving diffeomorphisms. The paper describes precise algebraic structure of groups π0S′(f,V) and some of their subgroups and quotients for a large class of smooth maps f:M→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 π0S′(f,V) are solvable and Bieberbach.