The cohomology of the projective -plane

Author:
William A. Thedford

Journal:
Proc. Amer. Math. Soc. **76** (1979), 327-332

MSC:
Primary 55R35; Secondary 57T25

DOI:
https://doi.org/10.1090/S0002-9939-1979-0537099-7

MathSciNet review:
537099

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Abstract | References | Similar Articles | Additional Information

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 . The main results of this paper are: Theorem 1 which states that where *N* is a truncated polynomial algebra over *k* and *S* is a trivial *k*-ideal, and Theorem 2 which considers the case and states that where is a truncated polynomial algebra on generators in even dimensions and *S* is an *A(p)*-sub-algebra of so that an *A(p)*-algebra structure can be induced on . These theorems extend results by A. Borel, W. Browder, M. Rothenberg, N. E. Steenrod, and E. Thomas.

**[1]**Armand Borel,*Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts*, Ann. of Math. (2)**57**(1953), 115–207 (French). MR**0051508**, https://doi.org/10.2307/1969728**[2]**William Browder and Emery Thomas,*On the projective plane of an 𝐻-space*, Illinois J. Math.**7**(1963), 492–502. MR**0151974****[3]**Allan Clark,*On 𝜋₃ of finite dimensional 𝐻-spaces*, Ann. of Math. (2)**78**(1963), 193–196. MR**0151975**, https://doi.org/10.2307/1970508**[4]**Albrecht Dold and Richard Lashof,*Principal quasi-fibrations and fibre homotopy equivalence of bundles.*, Illinois J. Math.**3**(1959), 285–305. MR**0101521****[5]**A. Dold and R. Thom,*Quasifaserungen und Unendliche Symmetrische Niedriger Dimension*, Fund. Math.**25**(1935), 427-440.**[6]**J. R. Hubbuck,*Generalized cohomology operations and H-spaces of low rank*(mimeographed).**[7]**John Milnor,*Construction of universal bundles. II*, Ann. of Math. (2)**63**(1956), 430–436. MR**0077932**, https://doi.org/10.2307/1970012**[8]**John W. Milnor and John C. Moore,*On the structure of Hopf algebras*, Ann. of Math. (2)**81**(1965), 211–264. MR**0174052**, https://doi.org/10.2307/1970615**[9]**M. Rothenberg and N. E. Steenrod,*The cohomology of classifying spaces of 𝐻-spaces*, Bull. Amer. Math. Soc.**71**(1965), 872–875. MR**0208596**, https://doi.org/10.1090/S0002-9904-1965-11420-3**[10]**J. Stasheff,*On a condition that a space is an H-space*. I, II, Trans. Amer. Math. Soc.**105**(1962), 126-175.**[11]**N. E. Steenrod,*Cohomology operations*, Lectures by N. E. STeenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. MR**0145525****[12]**Masahiro Sugawara,*A condition that a space is group-like*, Math. J. Okayama Univ.**7**(1957), 123–149. MR**0097066****[13]**W. A. Thedford,*On the A(p)*-*algebra structure of the Z(p)*-*cohomology of certain H-spaces*, Notices Amer. Math. Soc.**18**(1971), 223; Abstract #682-55-7.**[14]**-,*The Z(p)*-*cohomology of certain H-spaces*, Thesis, New Mexico State University, 1970.**[15]**Emery Thomas,*On functional cup-products and the transgression operator*, Arch. Math. (Basel)**12**(1961), 435–444. MR**0149485**, https://doi.org/10.1007/BF01650589

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0537099-7

Keywords:
*Z(p)*-cohomology of *H*-spaces,
Steenrod algebra,
projective plane of an *H*-space,
homotopy associativity

Article copyright:
© Copyright 1979
American Mathematical Society