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 Tr. Mosk. Mat. Obs., 2011, Volume 72, Issue 1, Pages 127–188 (Mi mmo14)

Topological applications of graded Frobenius $n$-homomorphisms

D. V. Gugnin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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.

Key words and phrases: graded algebra, graded $n$-homomorphism, Frobenius, Smith-Dold branched covering, cohomological length, $n$-valued topological group.

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English version:
Transactions of the Moscow Mathematical Society, 2011, 72, 97–142

Bibliographic databases:

UDC: 512.647+512.552+515.145.2
MSC: Primary 16W20, 17A42; Secondary 57M12
Revised: 05.01.2011

Citation: D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms”, Tr. Mosk. Mat. Obs., 72, no. 1, MCCME, Moscow, 2011, 127–188; Trans. Moscow Math. Soc., 72 (2011), 97–142

Citation in format AMSBIB
\Bibitem{Gug11} \by D.~V.~Gugnin \paper Topological applications of graded Frobenius $n$-homomorphisms \serial Tr. Mosk. Mat. Obs. \yr 2011 \vol 72 \issue 1 \pages 127--188 \publ MCCME \publaddr Moscow \mathnet{http://mi.mathnet.ru/mmo14} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3184814} \zmath{https://zbmath.org/?q=an:06026282} \elib{https://elibrary.ru/item.asp?id=21369339} \transl \jour Trans. Moscow Math. Soc. \yr 2011 \vol 72 \pages 97--142 \crossref{https://doi.org/10.1090/S0077-1554-2012-00191-5} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84959573947} 

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This publication is cited in the following articles:
1. D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms, II”, Trans. Moscow Math. Soc., 73 (2012), 167–182
2. D. V. Gugnin, “Lower Bounds for the Degree of a Branched Covering of a Manifold”, Math. Notes, 103:2 (2018), 187–195
3. D. V. Gugnin, “Razvetvlennye nakrytiya mnogoobrazii i $\boldsymbol{nH}$-prostranstva”, Funkts. analiz i ego pril., 53:2 (2019), 68–71
4. Johnson K.W., “Group Matrices, Group Determinants and Representation Theory the Mathematical Legacy of Frobenius Preface”: Johnson, KW, Group Matrices, Group Determinants and Representation Theory: the Mathematical Legacy of Frobenius, Lect. Notes Math., Lecture Notes in Mathematics, 2233, Springer International Publishing Ag, 2019, IX+
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