SOME HOMOTOPY CLASSIFICATION AND EXTENSION-THEOREMS FOR THE CLASS OF COMPOSITIONS OF ACYCLIC SET-VALUED MAPS

We study the homotopy properties of set-valued maps from a class conta ining, for example, the finite compositions of acyclic upper semi-cont inuous maps of finite-dimensional absolute neighbourhood retracts. We prove some generalizations of the classical Hopf theorems concerning h omotopy classification and the extension of set-valued maps. We show t hat under some mild conditions on spaces, each map from the considered class is homotopic within this class to a single-valued map. Therefor e the results of the paper yield answers to some old questions of the theory of set-valued maps. The main results rely on the proved theorem stating that a perfect surjection with acyclic fibres induces a bijec tive (co)transformation of the sets of homotopy classes and constituti ng an extension of the famous Vietoris theorem.