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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

   

 

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
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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$.


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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: http://dx.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