Cohomology of Products and Coproducts of Groups

Executive summary

Product means tensor; coproduct means sum

For a direct product the group ring factors as a tensor product, so Künneth applies and cohomology is a tensor product with a correction term. For a free product — the coproduct in the category of groups — the answer is entirely different: homology is the direct sum of the factors in positive degrees, with no interaction at all. The contrast is a good illustration that product and coproduct are genuinely different constructions.

Learning objectives

Apply Künneth to a direct product of groups.
State the homology of a free product.
Derive the Mayer–Vietoris sequence for an amalgamated product.
Explain why the two answers differ so sharply.

Section 01Direct products

ℤ[G × H] ≅ ℤ[G] ⊗_ℤ ℤ[H], so a tensor product of resolutions resolves the trivial module, and Künneth applies:

H_n(G × H) = ⊕_p+q=n H_p(G) ⊗ H_q(H) ⊕ Tor correction

Cohomology rings multiply

With field coefficients the cohomology ring of a direct product is the graded tensor product of the factors' rings, complete with Koszul signs. This makes elementary abelian groups computable: their cohomology is a polynomial or exterior algebra depending on the characteristic.

Section 02Free products

For a free product G * H:

H_n(G * H) ≅ H_n(G) ⊕ H_n(H) for n ≥ 1

with no tensor product and no correction. Topologically this is the statement that a wedge of classifying spaces is the classifying space of the free product, and the homology of a wedge is the direct sum in positive degrees.

Direct productInteraction

The factors interact: degrees add and torsion combines through Tor. The cohomology is genuinely larger than the sum of the parts.

Free productNo interaction

The factors do not interact at all above degree 0. Free products of free groups are free, and cohomological dimension is the maximum of the factors'.

Section 03Amalgamated products and Mayer–Vietoris

For an amalgamated free product G *_K H there is a Mayer–Vietoris sequence:

… → H_n(K) → H_n(G) ⊕ H_n(H) → H_n(G *_K H) → H_n−1(K) → …

Taking K trivial recovers the free product result. The sequence is the algebraic counterpart of the topological Mayer–Vietoris sequence for a space glued from two pieces along a common subspace.

Bass–Serre theory in the background

Amalgamated products and HNN extensions are exactly the groups acting on trees with specified stabilisers. The Mayer–Vietoris sequence is the algebraic shadow of that action, which is why geometric group theory and group cohomology are so closely linked.

ReferenceFrequently asked questions

Why does the free product have no correction term?

Because the corresponding topological construction is a wedge, and the homology of a wedge splits with no interaction. Algebraically, a free product's classifying space is built by gluing at a point, so no higher-dimensional interaction is created.

What is the cohomological dimension of a free product?

The maximum of the factors' dimensions, provided at least one is positive. This reflects that no new cohomology appears, in contrast to a direct product where dimensions add.

Does the Künneth correction ever matter for groups?

Yes — for a product of two groups each with torsion in homology, the Tor term contributes genuine extra classes. A product of two cyclic groups of even order is the smallest example.

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