Homological finiteness constrains group structure
The cohomological dimension of a group is the length of the shortest projective resolution of ℤ over its group ring. It is zero only for the trivial group, one exactly for free groups — the Stallings–Swan theorem — and infinite for any group with torsion. Finer finiteness conditions, FPn and Fn, ask that resolutions be finitely generated or that classifying spaces have finite skeleta, and they distinguish groups that dimension alone cannot.
Learning objectives
- Define cohomological and geometric dimension.
- State the Stallings–Swan theorem.
- Distinguish the FPn and Fn conditions.
- Explain why torsion forces infinite dimension.
Section 01Cohomological dimension
| Group | cd | Note |
|---|---|---|
| Trivial group | 0 | The only group with cd = 0 |
| Free group | 1 | Stallings–Swan: cd = 1 characterises free groups |
| ℤn | n | Classifying space is the n-torus |
| Surface group, genus ≥ 1 | 2 | Poincaré duality group |
| Finite non-trivial group | ∞ | Torsion forces periodic non-vanishing |
| SLn(ℤ) | ∞ | Contains torsion; torsion-free subgroups have finite cd |
A finite cyclic subgroup has periodic non-vanishing cohomology, and restriction is injective on the relevant primary part, so cohomology of the ambient group cannot vanish in high degrees. Hence any group with torsion has infinite cohomological dimension — no exceptions.
Section 02Stallings–Swan and geometric dimension
The geometric dimension gd(G) is the least dimension of a classifying space. Always cd ≤ gd, with equality except possibly when cd = 2 and gd = 3 — the Eilenberg–Ganea problem, still open.
- Suppose cd(G) ≤ 1, so ℤ has a projective resolution of length 1 over ℤ[G].
- Stallings proved the finitely generated case using ends of groups.
- Swan extended it to arbitrary groups.
- Conclusion: G is free. The converse is immediate, since a free group has a one-dimensional classifying space — a wedge of circles.
- So cd(G) ≤ 1 ⇔ G free, and freeness becomes a homological condition.
Section 03Finiteness properties
ℤ admits a projective resolution over ℤ[G] that is finitely generated in degrees up to n. FP1 is equivalent to finite generation of G.
G has a classifying space with finite n-skeleton. F1 is finite generation; F2 is finite presentability.
Fn implies FPn, and the converse holds given finite presentability. Bestvina–Brady groups separate the conditions: there are groups that are FP2 but not finitely presented.
Duality groups
Groups whose cohomology satisfies a duality with homology in complementary degree, generalising Poincaré duality. Surface groups and torsion-free arithmetic groups are examples.
Groups of type FP∞
Finitely generated in every degree. Includes many groups arising geometrically, and the condition behaves well under extensions.
Virtual duality groups
Have a finite-index subgroup that is a duality group. SLn(ℤ) is one, which is how its cohomology is studied despite having torsion.
ReferenceFrequently asked questions
Is the Eilenberg–Ganea problem still open?
Yes. It asks whether a group with cd = 2 must have gd = 2. No counterexample is known and no proof exists; it is one of the long-standing open problems in the homological theory of groups.
Why does virtual cohomological dimension help?
Because a group with torsion has infinite cd, but often has a torsion-free finite-index subgroup with finite cd. That number — the virtual cohomological dimension — is well defined by Serre's theorem and carries the useful information.
What distinguishes FP<sub>2</sub> from finite presentability?
FP2 is an algebraic condition on the relation module; finite presentability requires finitely many relations as a group. Bestvina–Brady constructed groups satisfying the first but not the second, settling a long-standing question.
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