ИСТИНА

We study the space F of smooth functions with prescribed local singularities of the A_k-types on a fxed smooth two-dimensional surface. We describe the homotopy type of this functional space F, endowed with the C∞-topology. We also describe the decomposition of F into the classes of topologically equivalent functions. Here, by a topological equivalence classes, we mean orbits of the action of the group of "left-right changings of coordinates" on F. It turns out that the (infinite-dimensional) space F has the homotopy type of a finite-dimensional manifold, consisting of nice blocks ("toric handles") and having a nice combinatorial structure.