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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Homologie de l'espace des sections d'un fibré


Author: Claude Legrand
Journal: Trans. Amer. Math. Soc. 297 (1986), 445-459
MSC: Primary 55R10; Secondary 55R20, 55S45, 55T05
DOI: https://doi.org/10.1090/S0002-9947-1986-0854077-7
MathSciNet review: 854077
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Abstract: For a fiber bundle with a finite cohomology dimension and $ 1$-connected base $ B$ and $ 1$-connected fiber $ F$, we obtain the homology of the section space by an $ {E^1}$-spectral sequence. In the "stable" range the $ {E^1}$-terms are the homology of a product of Eilenberg-Mac Lane space of type $ K({H^{p - i}}(B;{\pi _p}F),i)$ (the same as those of the $ {E^1}$-spectral sequences which converges to the homology of the functional space $ \operatorname{Hom} (B,F)$ [10]). The differential is the product of two operations: one appears in the $ {E^1}$-spectral sequence, which converges to the homology of $ \operatorname{Hom} (B,F)$; the second one is a "cup-product" determined by the fiber structure of the bundle. This spectral sequence is obtained by a Moore-Postnikov tower of the fiber, which generalizes Kahn's methods [9].


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DOI: https://doi.org/10.1090/S0002-9947-1986-0854077-7
Article copyright: © Copyright 1986 American Mathematical Society