Serre spectral sequence

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In mathematics, the Serre spectral sequence (sometimes Leray-Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is a basic tool of algebraic topology. It expresses, in the language of homological algebra the singular (co)homology of the total space E 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 (Serre's thesis).

Contents 1 Formulation 2 Cohomology spectral sequence 3 Homology spectral sequence 4 See also 5 References

[edit] Formulation

Let be a Serre fibration of topological spaces, and let F be "the" fiber. The result is expressed by means of a spectral sequence and associated standard notation. Without simplifying assumptions, the notation has to be read correctly.

[edit] Cohomology spectral sequence

The Serre cohomology spectral sequence is the following:

E₂^pq = H^p(B, H^q(F)) H^p+q(E).

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.

There is a multiplicative structure

coinciding on the E₂-term with 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.

[edit] Homology spectral sequence

Similarly to the cohomology spectral sequence, there is one for homology:

E²_pq = H_p(B, H_q(F)) H_p+q(E),

where the notations are dual to the ones above.

It is actually a special case of a more general spectral sequence, namely the Serre spectral sequence for fibrations of simplicial sets. If f is a fibration of simplicial sets (a Kan fibration), such that π₁(B), the first homotopy group of the simplicial set B, vanishes, there is a spectral sequence exactly as above. (Applying the functor which associates to any topological space its simplices to a fibration of topological spaces, one recovers the above sequence).

[edit] See also

• Gysin sequence
• Leray spectral sequence

[edit] References

The Serre spectral sequence is covered in most textbooks on algebraic topology, e.g.

• Allen Hatcher, The Serre spectral sequence
• Edwin Spanier, Algebraic topology, Springer

The case of simplicial sets is treated in

• P. Goerss, R. Jardine, Simplicial homotopy theory, Birkhäuser