Topological applications of graded Frobenius 𝑛-homomorphisms

Gugnin, D.

doi:10.1090/S0077-1554-2012-00191-5

Topological applications of graded Frobenius $n$-homomorphisms
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by D. V. Gugnin
Translated by: Alex Martsinkovsky

Trans. Moscow Math. Soc. 2011, 97-142

DOI: https://doi.org/10.1090/S0077-1554-2012-00191-5

Published electronically: January 12, 2012

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Abstract:

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.

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