Serre spectral sequence

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.
Cohomology spectral sequence.
Let formula_1 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 formula_3-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 formula_4, where formula_5 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 formula_5 to formula_9, "g" is defined using the coboundary map in the long exact sequence of the pair, and "h" is defined by restricting formula_4 to formula_11
There is a multiplicative structure
coinciding on the "E"2-term with (−1)"qs" times the cup product, and with respect to which the differentials formula_13 are (graded) derivations inducing the product on the formula_14-page from the one on the formula_15-page.
Homology spectral sequence.
Similarly to the cohomology spectral sequence, there is one for homology:
where the notations are dual to the ones above.
Example computations.
Hopf fibration.
Recall that the Hopf fibration is given by formula_17. The formula_3-page of the Leray–Serre Spectral sequence reads
The differential formula_20 goes formula_21 down and formula_22 right. Thus the only differential which is not necessarily is , because the rest have domain or codomain "0" (since they are on the "E"2-page). In particular, this sequence degenerates at "E"2 = "E"∞. The "E"3-page reads
The spectral sequence abuts to formula_24 i.e. formula_25 Evaluating at the interesting parts, we have formula_26 and formula_27 Knowing the cohomology of formula_28 both are zero, so the differential formula_29 is an isomorphism.
Sphere bundle on a complex projective variety.
Given a complex "n"-dimensional projective variety there is a canonical family of line bundles formula_30 for formula_31 coming from the embedding formula_32. This is given by the global sections formula_33 which send
If we construct a rank vector bundle formula_35 which is a finite whitney sum of vector bundles we can construct a sphere bundle formula_36 whose fibers are the spheres formula_37. Then, we can use the Serre spectral sequence along with the Euler class to compute the integral cohomology of . The formula_3-page is given by formula_39. We see that the only non-trivial differentials are given on the formula_40-page and are defined by cupping with the Euler class formula_41. In this case it is given by the top chern class of formula_35. For example, consider the vector bundle formula_43 for a K3 surface. Then, the spectral sequence reads as
The differential formula_45 for formula_46 is the square of the Lefschetz class. In this case, the only non-trivial differential is then
We can finish this computation by noting the only nontrivial cohomology groups are
Basic pathspace fibration.
We begin first with a basic example; consider the path space fibration 
We know the homology of the base and total space, so our intuition tells us that the Serre spectral sequence should be able to tell us the homology of the loop space. This is an example of a case where we can study the homology of a fibration by using the "E"∞ page (the homology of the total space) to control what can happen on the "E"2 page. So recall that 
Thus we know when "q" = 0, we are just looking at the regular integer valued homology groups "Hp"("S""n"+1) which has value formula_51 in degrees 0 and "n"+1 and value 0 everywhere else. However, since the path space is contractible, we know that by the time the sequence gets to "E"∞, everything becomes 0 except for the group at "p" = "q" = 0. The only way this can happen is if there is an isomorphism from formula_52 to another group. However, the only places a group can be nonzero are in the columns "p" = 0 or "p" = "n"+1 so this isomorphism must occur on the page "E""n"+1 with codomain formula_53 However, putting a formula_51 in this group means there must be a formula_51 at "H""n"+1("S""n"+1; "Hn"("F")). Inductively repeating this process shows that "Hi"(Ω"S""n"+1) has value formula_51 at integer multiples of "n" and 0 everywhere else.
Cohomology ring of complex projective space.
We compute the cohomology of formula_57 using the fibration:
Now, on the "E"2 page, in the 0,0 coordinate we have the identity of the ring. In the 0,1 coordinate, we have an element "i" that generates formula_59 However, we know that by the limit page, there can only be nontrivial generators in degree 2"n"+1 telling us that the generator "i" must transgress to some element "x" in the 2,0 coordinate. Now, this tells us that there must be an element "ix" in the 2,1 coordinate. We then see that "d"("ix") = "x"2 by the Leibniz rule telling us that the 4,0 coordinate must be "x"2 since there can be no nontrivial homology until degree 2"n"+1. Repeating this argument inductively until 2"n" + 1 gives "ixn" in coordinate 2"n",1 which must then be the only generator of formula_51 in that degree thus telling us that the 2"n" + 1,0 coordinate must be 0. Reading off the horizontal bottom row of the spectral sequence gives us the cohomology ring of formula_57 and it tells us that the answer is formula_62
In the case of infinite complex projective space, taking limits gives the answer formula_63
Fourth homotopy group of the three-sphere.
A more sophisticated application of the Serre spectral sequence is the computation formula_64 This particular example illustrates a systematic technique which one can use in order to deduce information about the higher homotopy groups of spheres. Consider the following fibration which is an isomorphism on formula_65
where formula_67 is an Eilenberg–MacLane space. We then further convert the map formula_68 to a fibration; it is general knowledge that the iterated fiber is the loop space of the base space so in our example we get that the fiber is formula_69 But we know that formula_70 Now we look at the cohomological Serre spectral sequence: we suppose we have a generator for the degree 3 cohomology of formula_71, called formula_72. Since there is nothing in degree 3 in the total cohomology, we know this must be killed by an isomorphism. But the only element that can map to it is the generator "a" of the cohomology ring of formula_73, so we have formula_74. Therefore by the cup product structure, the generator in degree 4, formula_75, maps to the generator formula_76 by multiplication by 2 and that the generator of cohomology in degree 6 maps to formula_77 by multiplication by 3, etc. In particular we find that formula_78 But now since we killed off the lower homotopy groups of "X" (i.e., the groups in degrees less than 4) by using the iterated fibration, we know that formula_79 by the Hurewicz theorem, telling us that formula_64
Corollary: formula_81
Proof: Take the long exact sequence of homotopy groups for the Hopf fibration formula_82.
References.
The Serre spectral sequence is covered in most textbooks on algebraic topology, e.g.
Also
An elegant construction is due to
The case of simplicial sets is treated in 