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Concept# Serre spectral sequence

Summary

In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The result is due to Jean-Pierre Serre in his doctoral dissertation.
Let be a Serre fibration of topological spaces, and let F be the (path-connected) fiber. The Serre cohomology spectral sequence is the following:
Here, at least under standard simplifying conditions, the coefficient group in the -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 , where 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 to , g is defined using the coboundary map in the long exact sequence of the pair, and h is defined by restricting to
There is a multiplicative structure
coinciding on the E2-term with (−1)qs times the cup product, and with respect to which the differentials are (graded) derivations inducing the product on the -page from the one on the -page.
Similarly to the cohomology spectral sequence, there is one for homology:
where the notations are dual to the ones above.

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