Serre Spectral Sequence - Cohomology Spectral Sequence

Cohomology Spectral Sequence

The Serre cohomology spectral sequence is the following:

Here, at least under standard simplifying conditions, the coefficient group in the E₂-term is the q-th integral cohomology group of F, and the outer group is the singular cohomology of B with coefficients in that group.

Strictly speaking, what is meant is cohomology with respect to the local coefficient system on B given by the cohomology of the various fibers. Assuming for example, that B is simply connected, this collapses to the usual cohomology. For a path connected base, all the different fibers are homotopy equivalent. In particular, their cohomology is isomorphic, so the choice of "the" fiber does not give any ambiguity.

The abutment means integral cohomology of the total space X.

This spectral sequence can be derived from an exact couple built out of the long exact sequences of the cohomology of the pair (X_p, X_p−1), where X_p is the restriction of the fibration over the p-skeleton of B. More precisely, using this notation,

,

f is defined by restricting each piece on X_p to X_p−1, g is defined using the coboundary map in the long exact sequence of the pair, and h is defined by restricting (X_p, X_p−1) to X_p.

There is a multiplicative structure

coinciding on the E₂-term with (−1)qs times the cup product, and with respect to which the differentials d_r are (graded) derivations inducing the product on the E_r+1-page from the one on the E_r-page.