The Five-Term Exact Sequence

Q: 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.

Q: 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.

Executive summary

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

0 → H¹(Q, A^N) →^inf H¹(G, A) →^res H¹(N, A)^Q →^tg H²(Q, A^N) → H²(G, A)

The three maps
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

Exactness is the useful content

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

AlgorithmReading the five-term sequence off the E<sub>2</sub> pagein: the LHS spectral sequence → out: the five-term exact sequence

The LHS spectral sequence has E₂^p,q = H^p(Q, H^q(N, A)) converging to H^p+q(G, A).
In total degree 1 only E₂^1,0 and E₂^0,1 contribute.
The only possibly non-zero differential in this range is d₂: E₂^0,1 → E₂^2,0. 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 E₂^1,0; restriction is the edge map to E₂^0,1.

Section 03Uses

Use

Hilbert 90 and Kummer theory

In Galois cohomology the sequence relates the cohomology of a subextension to the whole, underpinning descent arguments.

Use

Detecting non-split extensions

A non-vanishing transgression shows a class does not extend, which often proves an extension does not split.

Use

Computing H² of a quotient

When H¹(G) and H¹(N) are known, the sequence constrains H²(Q).

Use

Central extensions

For N central the sequence relates the Schur multiplier of Q to that of G.

Use

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.

Use

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.

NavigateContinue in this stream

Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review The Five-Term Exact Sequence. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat The Five-Term Exact Sequence as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—sequence, five-term, exact, spectral, section—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying The Five-Term Exact Sequence?