H1 and H2 determine the nilpotent quotients
A homomorphism inducing an isomorphism on H1 and a surjection on H2 induces isomorphisms on every quotient by the lower central series. That is Stallings' theorem, with Stammbach's variant supplying the nilpotent-group case. It converts an infinite family of group-theoretic questions into two homological ones, and it is the reason the Schur multiplier controls so much of nilpotent group theory.
Learning objectives
- Define the lower central series and nilpotency.
- State Stallings' theorem and Stammbach's variant.
- Explain the role of the Schur multiplier.
- Apply the results to a presentation question.
Section 01The lower central series
G is nilpotent of class c when γc+1G = 1. The quotients G/γnG are the nilpotent quotients, and γ2G is the commutator subgroup, so G/γ2G = H1(G, ℤ).
Each step of the lower central series is a central extension, and central extensions are classified by H². Building up the nilpotent quotients one step at a time is therefore a sequence of problems that H1 and H2 control.
Section 02The theorems
- Let f: G → K be a homomorphism.
- Suppose f induces an isomorphism H1(G) → H1(K).
- Suppose f induces a surjection H2(G) ↠ H2(K).
- Then f induces isomorphisms G/γnG → K/γnK for every finite n. The proof is an induction on n using the five-term exact sequence.
- It does NOT follow that f is an isomorphism, nor that the inverse limits agree.
Two groups can have all nilpotent quotients isomorphic and still differ. Parafree groups — non-free groups with the same nilpotent quotients as a free group — are the standard examples. The theorem is genuinely about the tower, not about the group.
Section 03Applications
Deficiency of presentations
The rank of the Schur multiplier bounds how far a presentation can be from balanced, constraining the number of relations required.
Detecting non-isomorphism
Two groups with different H2 cannot have isomorphic nilpotent completions in a way compatible with H1 — a computable obstruction.
Link theory
Milnor invariants of links are read from the nilpotent quotients of the link group, and Stallings' theorem is what makes them well defined.
Homology spheres
A group with the homology of the trivial group has trivial nilpotent quotients but need not be trivial — the source of exotic homology spheres.
Nilpotent completion
The inverse limit of the nilpotent quotients is determined by the hypotheses, though the groups themselves need not be.
Lower central series of free groups
The quotients are free abelian of ranks given by the necklace numbers, computed from the vanishing of H2 for free groups.
ReferenceFrequently asked questions
Why is H<sub>2</sub> only required to be surjective?
Because the induction uses the five-term exact sequence, where H2 appears at the end and only needs to hit enough to force the next quotient to match. Requiring an isomorphism would be a stronger hypothesis than the conclusion needs.
Does the theorem extend to the whole group?
Only under additional hypotheses, such as both groups being residually nilpotent and finitely generated with an appropriate completeness condition. Without them, parafree groups show the conclusion fails.
What is the connection with the Schur multiplier?
H2(G, ℤ) is the Schur multiplier, and Hopf's formula computes it from a presentation. So the hypotheses of Stallings' theorem are checkable directly from presentations of the two groups.
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