The cohomology of the projective 𝑛-plane

Thedford, William A.

doi:10.1090/S0002-9939-1979-0537099-7

The cohomology of the projective $n$-plane
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by William A. Thedford PDF

Proc. Amer. Math. Soc. 76 (1979), 327-332 Request permission

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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On the A(p)

algebra structure of the Z(p)

cohomology of certain H-spaces

18

The Z(p)

cohomology of certain H-spaces

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