Spectral Sequences — Mathematics Knowledge System

Overview

A spectral sequence is an algebraic machine that systematically computes homology or cohomology of a complex space by a sequence of successive approximations. Starting from initial data (the E_2 page), one iterates differentials to compute homology, converging at the E_infinity page to the graded pieces of a filtration of the target. Spectral sequences are the primary tool for computing cohomology of fiber bundles, Lie groups, classifying spaces, and algebraic varieties.

Intuition

Imagine trying to compute the cohomology of a complicated space by approximating it with simpler pieces. Each page E_r of the spectral sequence is a better approximation than the last. The differential on the r-th page connects classes that are 'r steps apart' in the filtration. When all differentials beyond some point vanish, the spectral sequence 'degenerates' and the E_infinity page directly gives the answer. For a fiber bundle F -> E -> B, the Serre spectral sequence lets you compute H*(E) from H*(B) and H*(F) by tracking how the fiber cohomology 'twists' over the base.

Formal Definition

Definition

A spectral sequence is a sequence of bigraded modules {E_r^{p,q}} for r >= 2 (or r >= 0, 1 in some conventions), together with differentials d_r: E_r^{p,q} -> E_r^{p+r, q-r+1} satisfying d_r^2 = 0, and isomorphisms E_{r+1}^{p,q} = ker(d_r)/im(d_r). The sequence converges to H^{p+q}(X) if the filtration is compatible.

d_r : E_r^{p,q} \to E_r^{p+r,\, q-r+1}, \quad d_r^2 = 0

Differential on page r of a cohomological spectral sequence

differential

E_{r+1}^{p,q} = \frac{\ker(d_r : E_r^{p,q} \to E_r^{p+r, q-r+1})}{\operatorname{im}(d_r : E_r^{p-r, q+r-1} \to E_r^{p,q})}

The (r+1)-th page is the homology of the r-th page

next-page

E_2^{p,q} = H^p(B; H^q(F)) \Rightarrow H^{p+q}(E)

Serre spectral sequence for a fibration F -> E -> B

serre-spectral-sequence

Notation

Notation	Meaning
$E_r^{p,q}$	The (p,q) entry on the r-th page of a spectral sequence
$d_r$	Differential on the r-th page, with bidegree (r, 1-r)
$E_\infty^{p,q}$	The infinity page, the stable value after all differentials vanish

Theorems

Theorem 1: Serre Spectral Sequence

Let F -> E -> B be a fibration with B simply connected and path-connected. There is a first-quadrant spectral sequence with E_2^{p,q} = H^p(B; H^q(F; R)) converging to H^{p+q}(E; R). The product structure is compatible with the cup products on E and on the base and fiber.

Theorem 2: Leray-Hirsch Theorem

Let p: E -> B be a fiber bundle with fiber F. If there exist cohomology classes c_1, ..., c_k in H*(E; R) that restrict to a basis of H*(F; R) on each fiber, then H*(E; R) is a free H*(B; R)-module with basis c_1, ..., c_k. In particular, H*(E; R) = H*(B; R) tensor H*(F; R) as R-modules.

Theorem 3: Degeneration at E_2

If E_2^{p,q} = 0 for all p > 0 or all q > 0 (the spectral sequence is concentrated in a row or column), then all differentials d_r = 0 for r >= 2. The spectral sequence degenerates at E_2 and gives H^n(E) = bigoplus_{p+q=n} E_2^{p,q}.

Worked Examples

1
Consider the Hopf fibration S^1 -> S^inf -> CP^inf (with S^inf contractible).
2
The Serre spectral sequence has E_2^{p,q} = H^p(CP^inf; Z) tensor H^q(S^1; Z).
3
Since S^inf is contractible, H^n(S^inf) = Z for n=0 and 0 otherwise, so the spectral sequence must converge to this.
4
H^q(S^1; Z) is Z for q=0,1 and 0 otherwise. The E_2 page has two non-zero rows: q=0 and q=1.
5
For convergence to the cohomology of a point (in high degrees), all classes must be killed by differentials. The d_2 differential must be an isomorphism H^p(CP^inf) -> H^{p+2}(CP^inf) in the relevant bidegrees.
6
This forces H^*(CP^inf; Z) = Z in all even degrees and 0 in odd degrees, i.e., H*(CP^inf; Z) = Z[x] with |x|=2.

✓ Answer

H*(CP^inf; Z) = Z[x] with |x| = 2, a polynomial algebra in one generator of degree 2.

Practice Problems

Hardproof writing

Using the Serre spectral sequence for the path-loop fibration Omega S^n -> P S^n -> S^n (where P S^n is contractible), compute H*(Omega S^n; Z) for n >= 2.

Hardfree response

Explain what it means for a spectral sequence to 'degenerate at E_2' and give an example.

Common Mistakes

Common Mistake

Knowing E_inf determines H*(E) completely.

The E_inf page gives the associated graded pieces of H*(E) with respect to a filtration. There can be non-trivial extension problems: knowing Gr H^n does not uniquely determine H^n if there is torsion. Extension problems must be resolved separately.

Common Mistake

The differentials on all pages are always zero after E_2.

Spectral sequences can have non-trivial differentials on pages E_3, E_4, etc. The Serre spectral sequence for the fibration S^3 -> S^7 -> S^4 (Hopf fibration) has a non-trivial d_4 differential.

Quiz

What does the E_2 page of the Serre spectral sequence for F -> E -> B equal?

H^p(B; H^q(F))H^p(F; H^q(B))H^p(E; H^q(F))H^{p+q}(F x B)

If a spectral sequence degenerates at E_2, what can you conclude?

H^n(E) = bigoplus_{p+q=n} E_2^{p,q} as groups (up to extension problems)H^n(E) = 0 for all nThe fiber is contractibleThe base space is simply connected

Historical Background

Spectral sequences were invented by Jean Leray in 1945 while he was a prisoner of war in Austria. Leray developed sheaf theory and spectral sequences as tools in analysis, which he concealed as abstract mathematics to avoid being put to work for the German war effort. Koszul later reformulated Leray's ideas in purely algebraic terms. Serre's 1951 thesis exploited spectral sequences (the Serre spectral sequence) to compute homotopy groups of spheres, a breakthrough that transformed algebraic topology.

1945
Leray invents spectral sequences and sheaf cohomology while a prisoner of war
Jean Leray
1947
Koszul gives an algebraic reformulation of Leray's spectral sequence
Jean-Louis Koszul
1951
Serre uses the Serre spectral sequence in his thesis to compute homotopy groups of spheres
Jean-Pierre Serre
1957
Grothendieck introduces the Grothendieck spectral sequence for derived functors
Alexander Grothendieck

Summary

A spectral sequence is a sequence of bigraded modules {E_r^{p,q}} with differentials d_r of bidegree (r, 1-r), where E_{r+1} is the homology of E_r.
The Serre spectral sequence for a fibration F -> E -> B has E_2^{p,q} = H^p(B; H^q(F)) and converges to H^{p+q}(E).
Degeneration at E_2 means E_2 = E_inf; the graded pieces of H*(E) are then just the E_2 entries, up to extension.
Spectral sequences are essential for computing homotopy and cohomology groups in situations where direct computation is intractable.
The Leray-Hirsch theorem is a useful criterion for degeneration when global cohomology classes restrict to fiber bases.

References

BookHatcher, A. Algebraic Topology. Cambridge University Press, 2002.
BookMcCleary, J. A User's Guide to Spectral Sequences, 2nd ed. Cambridge University Press, 2001.
BookBott, R. and Tu, L. Differential Forms in Algebraic Topology. Springer, 1982.