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 f : X → B be a Serre fibration of topological spaces, and let F be the fiber. The Serre cohomology spectral sequence is the following:

E_2^{p,q} = H^p(B, H^q(F)) \Rightarrow H^{p+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 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 (X_p, X_p−1), where X_p 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 X_p to X_p−1, g is defined using the coboundary map in the long exact sequence of the pair, and h is defined by restricting (X_p, X_p−1) to X_p.

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 d_r are (graded) derivations inducing the product on the E_r+1-page from the one on the E_r-page.

Homology spectral sequence

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

E^2_{p,q} = H_p(B, H_q(F)) \Rightarrow H_{p+q}(X).

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).

Example computations

Basic pathspace fibration

We begin first with a basic example; consider the path space fibration

\Omega \mathbf{S}^{n+1}\to \mathbf{PS}^{n+1}\to \mathbf{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 that

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

Thus we know when q = 0, we are just looking at the regular integer valued homology groups H_p(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 H_n+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; H_n(F)) = Z. However, putting a Z in this group means there must be a Z at H_n+1(Sⁿ⁺¹; H_n(F)). Inductively repeating this process shows that H_i(ΩSⁿ⁺¹) has value Z at integer multiples of n and 0 everywhere else.

Cohomology ring of complex projective space

We compute the cohomology of CPⁿ using the fibration:

\mathbf{S}^1\hookrightarrow \mathbf{S}^{2n+1}\to \mathbf{CP}^n

Now, on the E₂ 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 Z. However, we know that by the limit page, there can only be nontrivial generators in degree 2n+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) = 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 Z in that degree thus telling us that the 2n+1,0 coordinate must be 0. Reading off the horizontal bottom row of the spectral sequence gives us the cohomology ring of CPⁿ and it tells us that the answer is Z[x]/xⁿ⁺¹.

In the case of infinite complex projective space, taking limits gives the answer Z[x].

Fourth homotopy group of the three-sphere

A more sophisticated application of the Serre spectral sequence is the computation π₄(S³) = Z/2Z. 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 π₃

X\rightarrow \mathbf{S}^3\rightarrow K(\mathbf{Z},3).\

where K(π, n) is an Eilenberg-Maclane space. We then further convert the map X → S³ 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 cohomology, we know this must be killed by an isomorphism. But the only thing that can map to it is the generator a of the cohomology ring 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 particular we find that H₄(X) = Z/2Z. 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 π₄(S³) = Z/2Z.

Corollary: π₄(S²) = Z/2Z.

Proof: Take the long exact sequence of homotopy groups for the Hopf fibration S¹ → S³ → S².

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

An elegant construction is due to

A. Dress, Zur Spektralsequenz einer Faserung, Inventiones Mathematicae 3, p. 172-178 (1967)

The case of simplicial sets is treated in

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

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