On the equivalence of some spectral sequences for Serre fibrations

摘要

Several different constructions of a spectral sequence for a Serre fibration π: E → B over a compact simply connected manifold B are considered in this paper. Namely, we consider the spectral sequence for the minimal model (ΛV ? ΛW, d) of the fibration, along with the spectral sequences arising from the ?ech filtration in the complexes ?* (U, A*_(PL) (π~(-1) (U))) and C* (U, S* (π~(-1) (U))), where U = {U} is a covering of the base B. It is known that all these spectral sequences have the same terms E2*,* = H* ? 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 ≥ 2. 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 Ω*(E) and of the ?ech-de Rham complex. It is therefore established that all these constructions give the same spectral sequence, starting from the E _2 term.