Geodesic
Topological applications of graded Frobenius $n$-homomorphisms
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 72 (2011) no. 1, pp. 127-188.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MMO_2011_72_1_a4,
     author = {D. V. Gugnin},
     title = {Topological applications of graded {Frobenius} $n$-homomorphisms},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {127--188},
     publisher = {mathdoc},
     volume = {72},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMO_2011_72_1_a4/}
}
TY  - JOUR
AU  - D. V. Gugnin
TI  - Topological applications of graded Frobenius $n$-homomorphisms
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2011
SP  - 127
EP  - 188
VL  - 72
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MMO_2011_72_1_a4/
LA  - ru
ID  - MMO_2011_72_1_a4
ER  - 
%0 Journal Article
%A D. V. Gugnin
%T Topological applications of graded Frobenius $n$-homomorphisms
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2011
%P 127-188
%V 72
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MMO_2011_72_1_a4/
%G ru
%F MMO_2011_72_1_a4
D. V. Gugnin. Topological applications of graded Frobenius $n$-homomorphisms. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 72 (2011) no. 1, pp. 127-188. http://geodesic.mathdoc.fr/item/MMO_2011_72_1_a4/

[1] Bredon G., Vvedenie v teoriyu kompaktnykh grupp preobrazovanii, Nauka, M., 1980

[2] Bredon G., Teoriya puchkov, Nauka, M., 1988

[3] Bukhshtaber V.M., “Funktsionalnye uravneniya, assotsiirovannye s teoremami slozheniya dlya ellipticheskikh funktsii, i dvuznachnye algebraicheskie gruppy”, Uspekhi mat. nauk, 45:3 (1990), 185–186 | MR

[4] Bukhshtaber V.M., Novikov S.P., “Formalnye gruppy, stepennye sistemy i operatory Adamsa”, Matem. sb., 84:1 (1971), 81–118 | MR | Zbl

[5] Bukhshtaber V.M., Ris E.G., “Mnogoznachnye gruppy i $n$-algebry Khopfa”, Uspekhi mat. nauk, 51:4 (1996), 149–150 | MR

[6] Bukhshtaber V.M., Ris E.G., “Koltsa nepreryvnykh funktsii, simmetricheskie proizvedeniya i algebry Frobeniusa”, Uspekhi mat. nauk, 59:1 (2004), 125–144 | MR

[7] Bukhshtaber V.M., Ris E.G., “Konstruktivnoe dokazatelstvo obobschennogo izomorfizma Gelfanda”, Funkts. anal. pril., 35:4 (2001), 20–25 | MR

[8] Gugnin D.V., “Polinomialno zavisimye gomomorfizmy i $n$-gomomorfizmy Frobeniusa”, Trudy MIAN, 266, 2009, 64–96 | MR | Zbl

[9] Dold A., Lektsii po algebraicheskoi topologii, Mir, M., 1976

[10] Kuratovskii K., Topologiya, v. 1, Mir, M., 1966

[11] Panov T.E., “O strukture 2-algebry Khopfa v kogomologiyakh chetyrekhmernykh mnogoobrazii”, Uspekhi mat. nauk, 51:1 (1996), 161–162 | MR | Zbl

[12] Chernavskii A.V., “O konechnokratnykh otkrytykh otobrazheniyakh mnogoobrazii”, Matem. sb., 65 (1964), 357–369 | MR | Zbl

[13] Engelking R., Obschaya topologiya, Mir, M., 1986

[14] Alexander J.W., “Note on Riemann spaces”, Bull. Amer. Math. Soc., 26 (1920), 370–373 | DOI | MR

[15] Berstein I., Edmonds A.L., “The Degree and the Branch Set of a Branched Covering”, Invent. Matem., 45 (1978), 213–220 | DOI | MR | Zbl

[16] Buchstaber V.M., “$n$-Valued Groups: Theory and Applications”, Moscow Math. J., 6:1 (2006), 57–84 | MR | Zbl

[17] Buchstaber V.M., Rees E.G., “Multivalued groups, their representations and Hopf algebras”, Transform. Groups, 2:4 (1997), 325–349 | DOI | MR | Zbl

[18] Buchstaber V.M., Rees E.G., “Multivalued groups, $n$-Hopf algebras and $n$-ring homomorphisms”, Lie groups and Lie algebras, Math. Appl., 433, Kluwer Acad. Publ., Dordrecht, 1998, 85–107 | MR | Zbl

[19] Buchstaber V.M., Rees E.G., “The Gelfand map and symmetric products”, Selecta Math. (N.S.), 8:4 (2002), 523–535 | DOI | MR | Zbl

[20] Buchstaber V.M., Rees E.G., “Frobenius $n$-homomorphisms, transfers and branched coverings”, Math. Proc. Camb. Phil. Soc., 144:1 (2008), 1–12 | DOI | MR | Zbl

[21] Buchstaber V.M., Veselov A.P., “Integrable correspondences and algebraic representations of multivalued groups”, Internat. Math. Res. Notices, 8 (1996), 381–400 | DOI | MR

[22] Dold A., “Ramified coverings, orbit projections and symmetric powers”, Math. Proc. Camb. Phil. Soc., 99 (1986), 65–72 | DOI | MR | Zbl

[23] Dold A., “Homology of symmetric products and other functors of complexes”, Ann. of Math., 68 (1958), 54–80 | DOI | MR | Zbl

[24] Dragovic V., “Geometrization and Generalization of the Kowalevski top”, Comm. Math. Phys., 298:1 (2010), 37–64 | DOI | MR | Zbl

[25] Frobenius G., “Über Gruppencharaktere”, Sitzungsber. Preuß. Akad. Wiss. Berlin, 1896, 985–1021 | Zbl

[26] Frobenius G., “Über die Primfaktoren der Gruppendeterminante”, Sitzungsber. Preuß. Akad. Wiss. Berlin, 1896, 1343–1382 | Zbl

[27] Narasimhan R., Introduction to the theory of analytic spaces, Lecture Notes in Math., 25, Springer, Berlin, Heidelberg, New York, 1966 | MR | Zbl

[28] Smith L., “Transfer and ramified coverings”, Math. Proc. Camb. Phil. Soc., 93 (1983), 485–493 | DOI | MR | Zbl