Mishchenko K. M-functions on the noncompact surfaces.

There is a complete topologic classification of noncompact surfaces with a boundary in the dissertation. The topologic properties of noncompact surfaces with a boundary are studied. There are four orientability classes of noncompact surfaces with a boundary. It is proved, that main invariants of noncompact surfaces with a boundary are: the number of cuted points on the boundary, the number of glued up disks on the surface, reduced genus and orientability class. One of the most result of dissertation is complete topologic classification of m-functions on the noncompact surfaces, which has finite reduced genus, finite number of ideal boundaries and finite numbers of boundary components. Investigating of m-functions on the noncompact surfaces based on the introducing some m-function using unite of simple atoms and the rules of their gluing are obtained. An atom is neighborhood of critical point. So, any m-function may be represented by your own m-graph. It is proved, that two atoms are homeomorphic if andonly if their m-graphs are isomorphic. M-graph is set of edges and vertexes, which correspond to regular and critical points of m-function. M-graphs let us to give a complete topological classification of m-functions on the noncompact surfaces. The number of simple atoms for representation of any m-function on the noncompact surface of finite type are calculated. Keywords: noncompact surfaces, Morse functions on the manifolds, m-function on the surfaces, Reab's graphs, m-graphs, topological equivalence, isomorphism of graphs.