The dissertation is devoted to study homotopy properties of smooth functions on surfaces. Namely, we consider class F (M, P) of mappings, which consists of smooth mappings from the surface M into a circle or a line P, which take constant values on each connected component of the surface boundary and the critical points of which belong to the interior of surface and such that in the neighbourhood of each critical point they are smoothly equivalent to some homogeneous polynomials without multiple factors.
The main results that determine the scientific novelty of the dissertation:
– it is shown that for every map from the class F(B,P) of smooth maps on the Mobius strip B, there is a unique critical level that splits B into the union of a cylinder and 2-disks (such a level is called special).
-- for all mappings from F(B,P) the fundamental groups of their orbits are calculated in case where the actions of the stabilizers of these mappings are trivial on the connected components to the corresponding special critical levels;
-- proved that for an arbitrary mapping from the class F(M, P) on a connected orientable compact surface M and for an arbitrary diffeomorphism that leaves invariant each regular component of the level set of this mapping and changes its orientation, the square of this diffeomorphism is isotopic to an identical mapping with preserving the map (this statement is a homotopic and foliated analog of the property of <<rigidity>> for orientation-changing linear motions of the plane, which states that each such movement has an order of 2);
-- there was considered the isomorphism class of groups T, which is generated by direct products and certain types of wreath products, which contains fundamental groups of orbits of all functions from the class F(M, R) on oriented surfaces except for the 2-sphere. The following results have been proven for it:
- realization theorems are obtained for groups from the class T as funda-mental groups of orbits of functions from the class F(M, P) on surfaces other than the 2-sphere and 2-torus, in particular under certain restrictions on the behavior of functions at the boundary;
- also obtained realization theorems for groups from the class T as funda-mental groups of orbits of functions of the class F(T^2, R) on the 2-torus T^2;
- calculated the center Z(G) and the quotient group by the commutant G/[G,G] for each group G of the class T and showed that they are free Abelian groups of the same rank b_1. In particular, if G is the fundamental group of the orbit of some function from F(M,R), then b_1 is the first Betti number of this orbit, that is, the rank of the first homology group.
