Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

Request Permissions   Purchase Content 
 

 

Topological applications of graded Frobenius $ n$-homomorphisms, II


Author: D. V. Gugnin
Translated by: A. Martsinkovsky
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 167-182
MSC (2010): Primary 13A02, 16T05, 55P45, 57N65
Published electronically: January 24, 2013
MathSciNet review: 3184973
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper strengthens Theorems 3.1.7 and 3.2.4 in Topological applications of graded Frobenius n-homomorphisms, by D. V,Gugnin, Tr. Mosk. Mat. Obs. 72 (2011), no. 1, 127-188; English transl., Trans. Moscow Math. Soc. 72 (2011), no. 1, 97-142. The improved version of Theorem 3.1.7 allows us to use integral techniques when working with rational cohomology algebras of $ nH$-spaces. We introduce a rather broad class of even-dimensional manifolds  $ \mathcal {M}$ and, using integrality conditions, we show that those manifolds do not admit a 2-valued multiplication with identity. In particular, we show that complex projective spaces $ \mathbb{C}P^m, m\ge 2,$ are not $ 2H$-spaces. This fact has only been known for  $ \mathbb{C}P^2$.


References [Enhancements On Off] (What's this?)

  • 1. Victor M. Buchstaber, 𝑛-valued groups: theory and applications, Mosc. Math. J. 6 (2006), no. 1, 57–84, 221 (English, with English and Russian summaries). MR 2265947
  • 2. V. M. Buchstaber and E. G. Rees, Multivalued groups, their representations and Hopf algebras, Transform. Groups 2 (1997), no. 4, 325–349. MR 1486035, 10.1007/BF01234539
  • 3. V. M. Bukhshtaber, Functional equations that are associated with addition theorems for elliptic functions, and two-valued algebraic groups, Uspekhi Mat. Nauk 45 (1990), no. 3(273), 185–186 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 3, 213–215. MR 1071939, 10.1070/RM1990v045n03ABEH002361
  • 4. D. V. Gugnin, Topological applications of graded Frobenius $ n$-homomorphisms, Tr. Mosk. Mat. Obs. 72 (2011), no. 1, 127-188; English transl., Trans. Moscow Math. Soc. 72 (2011), no. 1, 97-142.
  • 5. D. V. Gugnin, Polynomially dependent homomorphisms and Frobenius 𝑛-homomorphisms, Tr. Mat. Inst. Steklova 266 (2009), no. Geometriya, Topologiya i Matematicheskaya Fizika. II, 64–96 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 266 (2009), no. 1, 59–90. MR 2603261, 10.1134/S0081543809030043
  • 6. Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119–221 (French). MR 0102537
  • 7. I. G. Macdonald, The Poincaré polynomial of a symmetric product, Proc. Cambridge Philos. Soc. 58 (1962), 563–568. MR 0143204
  • 8. I. G. Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962), 319–343. MR 0151460
  • 9. M. M. Postnikov, Lektsii po algebraicheskoi topologii, “Nauka”, Moscow, 1984 (Russian). Osnovy teorii gomotopii. [Elements of homotopy theory]. MR 776974

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 13A02, 16T05, 55P45, 57N65

Retrieve articles in all journals with MSC (2010): 13A02, 16T05, 55P45, 57N65


Additional Information

D. V. Gugnin
Affiliation: Chair of Higher Geometry and Topology, Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia
Email: dmitry-gugnin@yandex.ru

DOI: https://doi.org/10.1090/S0077-1554-2013-00201-0
Keywords: Symmetric power, graded algebra, $n$-valued topological group, graded Frobenius $n$-homomorphism, $n$-Hopf algebra
Published electronically: January 24, 2013
Additional Notes: Supported by the RFFI grants 11-01-00694-a and 12-01-92104-YaF-a, and the RF Government Grant RF no.2010-220-01-077, Contract 11.G34.31.0005.
Article copyright: © Copyright 2013 American Mathematical Society