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.
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
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.
More precisely, using this notation, f is defined by restricting each piece on
There is a multiplicative structure coinciding on the E2-term with (−1)qs times the cup product, and with respect to which the differentials
Similarly to the cohomology spectral sequence, there is one for homology: where the notations are dual to the ones above, in particular the differential on the kth page is a map
Given a complex n-dimensional projective variety X there is a canonical family of line bundles
which send If we construct a rank r vector bundle
Then, we can use the Serre spectral sequence along with the Euler class to compute the integral cohomology of S. The
In this case, the only non-trivial differential is then We can finish this computation by noting the only nontrivial cohomology groups are 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.
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.
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 En+1 with codomain
Inductively repeating this process shows that Hi(ΩSn+1) has value
using the fibration: Now, on the E2 page, in the 0,0 coordinate we have the identity of the ring.
We then see that d(ix) = x2 by the Leibniz rule telling us that the 4,0 coordinate must be x2 since there can be no nontrivial homology until degree 2n+1.
Repeating this argument inductively until 2n + 1 gives ixn in coordinate 2n,1 which must then be the only generator of
Reading off the horizontal bottom row of the spectral sequence gives us the cohomology ring of
In the case of infinite complex projective space, taking limits gives the answer
A more sophisticated application of the Serre spectral sequence is the computation
This particular example illustrates a systematic technique which one can use in order to deduce information about the higher homotopy groups of spheres.
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
Therefore by the cup product structure, the generator in degree 4,
by multiplication by 2 and that the generator of cohomology in degree 6 maps to
Proof: Take the long exact sequence of homotopy groups for the Hopf fibration
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