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Transactions of the Moscow Mathematical Society

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Topological applications of graded Frobenius $n$-homomorphisms

Author: D. V. Gugnin
Translated by: Alex Martsinkovsky
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 72 (2011), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2011, 97-142
MSC (2010): Primary 17A42; Secondary 57M12
Published electronically: January 12, 2012
MathSciNet review: 3184814
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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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Additional Information

D. V. Gugnin
Affiliation: Mechanics and Mathematics Department, Moscow State University, Moscow 11991, Russia

Keywords: Graded algebra, graded $n$-homomorphism, Frobenius, Smith–Dold branched covering, cohomological length, $n$-valued topological group
Published electronically: January 12, 2012
Additional Notes: Supported by the RFFI grants 10-01-92102-YaF-a and 11-01-00694-a, President’s Grant for leading scientific schools, Project NSh-5413.2010.1, and the Government Grant 2010-220-01-077, Contract 11.G34.31.0005.
Article copyright: © Copyright 2012 American Mathematical Society