Homologie de l’espace des sections d’un fibré

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].