Products of objects, tensor products of invariants
The Künneth formulas convert a product of objects into a tensor product of invariants, up to a correction. In topology this computes the homology of a product space; in group cohomology it computes the cohomology of a direct product of groups; in algebra it relates the homology of a tensor product of algebras to the factors. In each case the correction term is where torsion interacts, and where the clean multiplicative picture breaks.
Learning objectives
- Compute the homology of a product from the factors.
- Apply the formula to the cohomology of a direct product of groups.
- Relate the Künneth isomorphism to the cup product.
- Identify the cases where the correction is unavoidable.
Section 01Products of spaces
For spaces X and Y the singular chain complex of the product is chain equivalent to the tensor product of the chain complexes, so Künneth applies directly.
| Product | Homology |
|---|---|
| S1 × S1 | ℤ, ℤ², ℤ in degrees 0, 1, 2 — no Tor term, homology is free |
| ℝP2 × ℝP2 | Tor term is non-zero, since H1 = ℤ/2 in each factor |
| X × point | H*(X) — the formula degenerates correctly |
| Any product, field coefficients | Tensor product of the graded vector spaces |
Tor1(ℤ/2, ℤ/2) = ℤ/2, so a product of two spaces each with 2-torsion acquires extra homology that is not visible in the tensor product of the homologies. Real projective spaces are the standard example.
Section 02Direct products of groups
For groups G and H, the group ring of the direct product is the tensor product of the group rings, so a tensor product of resolutions resolves the trivial module and Künneth applies.
A coproduct of groups — a free product — behaves completely differently: its homology is the direct sum of the factors' homology in positive degrees, with no tensor product and no correction. Direct and free products are opposite constructions, and the cohomology reflects that.
Section 03Ring structure
With field coefficients the Künneth isomorphism is a ring isomorphism onto the graded tensor product, where multiplication carries the Koszul sign:
Omitting it makes the product fail to be associative or graded-commutative. Every graded structure in this subject carries such signs, and they are the commonest source of arithmetic errors in explicit cohomology ring computations.
The cup product on the cohomology of a space is compatible with the Künneth isomorphism, which is how cohomology rings of products are computed — and it is the principal reason cohomology is often preferred to homology despite the extra correction term.
ReferenceFrequently asked questions
Why does cohomology have a ring structure but homology not?
Because the diagonal map of a space induces a map from cohomology of the product to cohomology of the space, and Künneth identifies the former with a tensor product. In homology the diagonal goes the wrong way, so no product is induced without extra structure.
Does the correction term affect the ring structure?
It can. With field coefficients there is no correction and the isomorphism is multiplicative. Integrally, the extension recorded by the short exact sequence means the ring structure is not determined by the factors alone.
Is there a Künneth formula for group cohomology of a semidirect product?
Not directly — a semidirect product is not a direct product, so the group ring does not factor. The appropriate tool is the Lyndon–Hochschild–Serre spectral sequence, which reduces to Künneth in the split trivial-action case.
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