Serre spectral sequence

In mathematics, the Serre spectral sequence (sometimes Leray-Serre spectral sequence to acknowledge earlier work of Jean Leray) is a basic tool of algebraic topology. It expresses 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").

Let $f\; :\; E\; ightarrow\; B$ be a Serre fibration of topological spaces, and let "F" be "the" fiber.

Cohomology spectral sequence

This is the following spectral sequence:

:"E"₂^"pq" = "H"^"p"("B", "H"^"q"("F")) $Rightarrow$ "H"^"p+q"("E"),

where the coefficient group in the "E"₂-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, it means 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, i.e. $pi\_1(B)\; =\; 0$, this collapses to the usual cohomology). Note that for a path connected base, 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.

Homology spectral sequence

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

:"E"²_"pq" = "H"_"p"("B", "H"_"q"("F")) $Rightarrow$ "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 $pi\_1(B)$, 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).

References

The Serre spectral sequence is covered in most textbooks on algebraic topology, e.g.
* A. Hatcher, [http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html The Serre spectral sequence]
* E. Spanier, Algebraic topology, SpringerThe case of simplicial sets is treated in
* P. Goerss, R. Jardine, Simplicial homotopy theory, Birkhäuser