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On the complexes of Hirsch and Eilenberǵ–Moore

Published online by Cambridge University Press:  24 October 2008

W. D. Barcus
W. D. Barcus
Affiliation:
Department of Mathematics, State University of New York, Stony Brook, New York 11790, U.S.A.
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Extract

Both Hirsch (cf. (5), (3)) and Eilenberg and Moore (cf. (7)) have described filtered complexes for the cohomology of a fibre space, under different hypotheses. Provided both complexes are defined, we shall give an explicit filtration-preserving map of the first into the second which induces homology isomorphisms. In section 3 we shall use this to transfer the action of the group in a principal bundle from the Hirsch complex (as computed by the author in (1)) to that of Eilenberg and Moore. The latter complex has certain advantages, for example, it is natural, and admits a cup-product structure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 4 , October 1968 , pp. 953 - 960
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Barcus, W. D.Principal bundles and Hirsch's spectral sequence. Quart. J. Math., Oxford Ser. 17 (1966), 321–34.CrossRefGoogle Scholar
(2)Cockcroft, W. H.The cohomology groups of a fibre space with fibre a space of type K(π, n). II. Trans. Amer. Math. Soc. 91 (1959), 505–24.Google Scholar
(3)Cockcroft, W. H.On a theorem of A. Borel. Trans. Amer. Math. Soc. 98 (1961), 255–62.CrossRefGoogle Scholar
(4)Hirsch, G.Quelques propriétés des produits de Steenrod, C. R. Acad. Sci., Paris 241 (1955), 923–5.Google Scholar
(5)Hirsch, G.Sur les groupes d'homologie des espaces fibrés. Bull. Soc. Math. Belg. 6 (1953), 79–96.Google Scholar
(6)May, J. P. The algebraic Eilenberg–Moore spectral sequence (to appear).Google Scholar
(7)Smith, L. Homological algebra and the Eilenberg–Moore spectral sequence (to appear).Google Scholar