On the equivalence of some spectral sequences for Serre fibrations

A. Yu. Onishchenko ; F. Yu. Popelenskii

Sbornik. Mathematics, Tome 202 (2011) no. 4, pp. 547-570

Résumé

Several different constructions of a spectral sequence for a Serre fibration $\pi\colon E \to B$ over a compact simply connected manifold $B$ are considered in this paper. Namely, we consider the spectral sequence for the minimal model $(\Lambda V\otimes \Lambda W,d)$ of the fibration, along with the spectral sequences arising from the Čech filtration in the complexes $\check{C}^*(\mathscr{U}, A_{PL}^*(\pi^{-1}(U)))$ and $\check{C}^*(\mathscr{U}, S^*(\pi^{-1}(U)))$, where $\mathscr{U}=\{U\}$ is a covering of the base $B$. It is known that all these spectral sequences have the same terms $E_2^{*,*}=H^*(X)\otimes H^*(F)$ and converge to the cohomology of the total space $E$. A new natural isomorphism of these spectral sequences is constructed in every term $E_r$ with $r\ge2$. It is also proved that in the case of a smooth locally trivial fibration these spectral sequences are isomorphic to the spectral sequences of the complex of smooth forms $\Omega^*(E)$ and of the Čech-de Rham complex. It is therefore established that all these constructions give the same spectral sequence, starting from the $E_2$ term. Bibliography: 9 titles.

Détail

Keywords: spectral sequences, Čech-de Rham complex, minimal model.
Mots-clés : Serre fibration