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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

The cohomology of the projective $ n$-plane


Author: William A. Thedford
Journal: Proc. Amer. Math. Soc. 76 (1979), 327-332
MSC: Primary 55R35; Secondary 57T25
MathSciNet review: 537099
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Abstract: An H-space is a topological space with a continuous multiplication and an identity element. In this paper X has the homotopy type of a countable CW-complex with integral cohomology of finite type and primitively generated k-cohomology, k a field. The projective n-plane of X is denoted $ XP(n)$. The main results of this paper are: Theorem 1 which states that $ {H^\ast}(XP(n)) = N \oplus S$ where N is a truncated polynomial algebra over k and S is a trivial k-ideal, and Theorem 2 which considers the case $ k = Z(p)$ and states that $ {H^\ast}(XP(n)) = \hat N \oplus \hat S$ where $ \hat N$ is a truncated polynomial algebra on generators in even dimensions and S is an A(p)-sub-algebra of $ {H^\ast}(XP(n))$ so that an A(p)-algebra structure can be induced on $ \hat N$. These theorems extend results by A. Borel, W. Browder, M. Rothenberg, N. E. Steenrod, and E. Thomas.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0537099-7
PII: S 0002-9939(1979)0537099-7
Keywords: Z(p)-cohomology of H-spaces, Steenrod algebra, projective plane of an H-space, homotopy associativity
Article copyright: © Copyright 1979 American Mathematical Society