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ISSN 1547-738X(e) ISSN 0077-1554(p)

Topological applications of graded Frobenius -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
Posted: January 12, 2012
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Abstract: This paper generalizes the theory of Frobenius -homomorphisms, as expounded by V. M. Buchstaber and E. G. Rees, to graded algebras, and applies the new algebraic technique of graded Frobenius -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 -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 -homomorphism.

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