User:LkNsngth/serre spectral sequence sandbox

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 inalgebraic 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.

Formulation
Let $$f : X \rightarrow B$$ 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.

Cohomology spectral sequence
The Serre cohomology spectral sequence is the following:


 * E2pq = Hp(B, Hq(F))$$\Rightarrow$$ Hp+q(X).

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 ofB 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 (Xp, Xp-1), whereXp is the restriction of the fibration over the p-skeleton of B. More precisely, using this notation,


 * $$A = \bigoplus_{p,q} H^q(X_p)$$, $$E_1^{p,q} = C = \bigoplus_{p,q} H^q(X_p, X_{p-1})$$,

f is defined by restricting each piece on Xp to Xp-1, g is defined using the coboundary map in the LES of the pair, and h is defined by restricting (Xp, Xp-1) to Xp.

There is a multiplicative structure


 * $$E_r^{p,q} \times E_r^{s,t} \to E_r^{p+s,q+t},$$

coinciding on the E2-term with (-1)qs times the cup product, and with respect to which the differentials dr are (graded) derivations inducing the product on the Er+1-page from the one on the Er-page.

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


 * E2pq = Hp(B, Hq(F))$$\Rightarrow$$ Hp+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).

A Basic Pathspace Fibration
We begin first with a basic example; consider the path space fibration

$$ \Omega S^{n+1}\rightarrow PS^{n+1}\rightarrow S^{n+1} $$

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 E2 page. So recall the E2p,q page is given by

$$ E^2_{p,q} = H_p(S^{n+1}; H_q(\Omega S^{n+1})) $$

Thus we know when q=0, we are just looking at the regular integer valued homology groups Hp(Sn+1) which has value Z 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 Hn+1(Sn+1; H0(F)) = Z 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 En+1 with codomain H0(Sn+1;Hn(F))=Z However, putting a Z in this group means there must be a Z at Hn+1(Sn+1; Hn(F)). Inductively repeating this process shows that Hi(Ω Sn+1) has value Z at integer multiples of n and 0 everywhere else.

The Fourth Homotopy Group of the Three Sphere
A more sophisticated application of the Serre spectral sequence is the computation π4S3=Z/2. This particular example illustrates a systematic technique which one can use in order to deduce information about the higher homotopy groups of spheres. We consider the following fibration which is an isomorphism on π3

$$X\rightarrow S^3\rightarrow K(\mathbb{Z},3)$$

where K(π, n) is an Eilenberg-Maclane space. We then further convert the map $$ X\rightarrow S^3 $$ 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 ΩK(Z,3)=K(Z,2). But we know that K(Z,2)=CP∞. Now we look at the cohomological Serre spectral sequence: we suppose we have a generator for the degree 3 cohomology of S3 called i. Since there is nothing in degree 3 in the total homology, we know this must be killed by an isomorphism. But the only thing that can map to it is the generator a of the homology of CP∞ so we have d(a)=i. Therefore by the cup product structure, the generator in degree 4, a2 maps to the generator ia by multiplication by 2 and that the generator of cohomology in degree 6 maps to ia2 by multiplication by 3 etc. In partiucular we find that H4 X = Z/2. But now since we killed off lower homotopy groups of X (i.e. groups in dimension less than 4) by using the iterated fibration, we know that H4 X = π4 X by the Hurewicz theorem telling us that π4S3=Z/2.