Functions on surfaces and incompressible subsurfaces

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Let M be a smooth connected compact surface and P be either a real line or a circle. This paper proceeds the study of the stabilizers and orbits of smooth functions on M with respect to the right action of the group of diffeomorphisms of M. A large class of smooth maps f:M→ P with isolated singularities is considered and it is shown that the general problem of calculation of the fundamental group of the orbit of f reduces to the case when the Euler characteristic of M is non-negative. For the proof of main result incompressible subsurfaces and cellular automorphisms of surfaces are investigated.