The first useful consequence of a spectral sequence
For a normal subgroup N of G with quotient Q, there is an exact sequence relating the cohomology of Q, of G and of N in low degrees. It is the edge of the Lyndon–Hochschild–Serre spectral sequence, but it can be stated and used with no spectral sequence machinery, and it is the standard first tool for relating a group's cohomology to that of a normal subgroup and quotient.
Learning objectives
- State the five-term exact sequence in cohomology and homology.
- Identify inflation, restriction and transgression.
- Derive it as the edge of a spectral sequence.
- Apply it to a concrete computation.
Section 01The sequence
| Map | Direction | Meaning |
|---|---|---|
| Inflation | From the quotient to the whole group | Pull back a cocycle along G ↠ Q — a cocycle constant on N-cosets |
| Restriction | From the whole group to the subgroup | Restrict a cocycle to N; the image lands in the Q-invariants |
| Transgression | From H1(N) to H2(Q) | The differential d2 of the spectral sequence — the obstruction to an N-cocycle extending to G |
A class on N extends to G exactly when its transgression vanishes, and two extensions differ by an inflated class. This converts an extension question into a cohomological computation on the quotient.
Section 02Derivation from the spectral sequence
- The LHS spectral sequence has E2p,q = Hp(Q, Hq(N, A)) converging to Hp+q(G, A).
- In total degree 1 only E21,0 and E20,1 contribute.
- The only possibly non-zero differential in this range is d2: E20,1 → E22,0. That differential is the transgression.
- Assembling the filtration in degrees 1 and 2 gives exactly the five-term sequence.
- Inflation is the edge map from E21,0; restriction is the edge map to E20,1.
Section 03Uses
Hilbert 90 and Kummer theory
In Galois cohomology the sequence relates the cohomology of a subextension to the whole, underpinning descent arguments.
Detecting non-split extensions
A non-vanishing transgression shows a class does not extend, which often proves an extension does not split.
Computing H² of a quotient
When H1(G) and H1(N) are known, the sequence constrains H²(Q).
Central extensions
For N central the sequence relates the Schur multiplier of Q to that of G.
Homology version
The dual sequence runs H2(G) → H2(Q) → (N/[G,N]) → H1(G) → H1(Q) → 0, giving Hopf's formula as a special case.
Inductive computation
For a group with a normal series, iterating the sequence computes low-degree cohomology step by step.
ReferenceFrequently asked questions
Why does the sequence stop at H²?
Because beyond total degree 2 further differentials and filtration steps intervene, so no short exact statement is available. Extending further requires the full spectral sequence.
Is inflation always injective?
In the five-term sequence, yes at H1 — that is part of the exactness statement. In higher degrees inflation need not be injective, which is one of the reasons the sequence cannot be continued naively.
What exactly is transgression?
The differential d2 of the spectral sequence, from the row of N-cohomology to the row of Q-cohomology two columns along. Concretely it takes an N-invariant class and measures the obstruction to extending a representing cocycle over G.
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