Topological applications of graded Frobenius $n$-homomorphisms

This paper generalizes the theory of Frobenius $n$-homomorphisms, as expounded by V. M. Buchstaber and E. G. Rees, to graded algebras, and applies the new algebraic technique of graded Frobenius $n$-homomorphisms to two topological problems. The first problem is to find estimates on the cohomological length of the base and of the total space of a wide class of branched coverings of topological spaces, called the Smith-Dold branched coverings. This class of branched coverings contains, in particular, unbranched finite-sheeted coverings and the usual finite-sheeted branched coverings from the theory of smooth manifolds. The second problem concerns a description of cohomology and fundamental groups of $n$-valued topological groups. The main tool there is a generalization of the notion of a graded Hopf algebra, based on the notion of a graded Frobenius $n$-homomorphism.