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Topological applications of graded Frobenius $$n$$-homomorphisms. (English. Russian original) Zbl 1254.54030
Trans. Mosc. Math. Soc. 2011, No. 1, 97-142 (2011); translation from Tr. Mosk. Mat. O.-va 2011, 46 p. (2011).
This paper is divided into three parts. In the first part the author generalizes the work of V. M. Buchstaber and E. G. Rees on Frobenius $$n$$-homomorphisms to the case of graded algebras. In the second part the author applies the new algebraic technique of Frobenius $$n$$-homomorphisms to find estimates on the cohomological length of the base and 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. In the third part the author attacks a problem concerning the description of the cohomology 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.

##### MSC:
 54C40 Algebraic properties of function spaces in general topology 17A42 Other $$n$$-ary compositions $$(n \ge 3)$$ 57M12 Low-dimensional topology of special (e.g., branched) coverings
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