Functions on surfaces and incompressible subsurfaces - Maksymenko, S. (Methods of Functional Analysis and Topology, 2010)
Let M be a smooth connected compact surface, P be either a real line R or a circle S^1. Then we have a natural right action of the group D(M) of diffeomorphisms of M on C^∞(M,P). For fC^∞(M,P) denote respectively by S(f) and O(f) its stabilizer and orbit with respect to this action. Recently, for a large class of smooth maps f:M→ P the author calculated the homotopy types of the connected components of S(f) and O(f). It turned out that except for few cases the identity component of S(f) is contractible, π_i O(f)=π_i M for i≥3, and π_2 O(f)=0, while π_1 O(f) it only proved to be a finite extension of π_1D_IdMZ^l for some l≥0. In this note it is shown that if χ(M)