the Künneth formula
/ KEUN-net /
If you know the homology of two spaces, what is the homology of their product? The answer is pleasingly close to 'multiply them' — the homology of a product is built from the homologies of the factors. The Künneth formula makes this precise, and it is the reason the homology of tori, products of spheres, and many test spaces can be written down instantly.
Over a field F the formula is clean: H_n(X cross Y; F) is the direct sum over all p + q = n of H_p(X; F) tensor H_q(Y; F). So the Betti numbers multiply in the sense of generating functions: the Poincare polynomial of a product is the product of the Poincare polynomials. Over the integers there is an extra correction by a torsion term: H_n(X cross Y; Z) sits in a short exact sequence whose left part is the direct sum of H_p(X) tensor H_q(Y) over p + q = n, and whose right part is the direct sum of Tor(H_p(X), H_q(Y)) over p + q = n - 1. The sequence splits non-naturally, so over Z the homology of a product is tensor products plus a shifted Tor correction.
The everyday use is computing products. The n-torus is a product of n circles, so its homology is the n-fold tensor power of the homology of a circle; the result is that H_k(T^n) is free abelian of rank 'n choose k', the binomial coefficients. There is a parallel Künneth formula in cohomology, where over a field it is even multiplicative as rings: the cohomology ring of a product is the (graded) tensor product of the cohomology rings, so the cup-product structure of a product is determined by the factors.
The honest fine print. Over a field there is no Tor term and life is simple; over Z you must not forget the torsion correction — claiming H_n(X cross Y; Z) is just the tensor sum is wrong whenever either factor has torsion. The splitting is again non-natural. And the clean field statement, like the universal coefficient theorem it generalizes, relies on the coefficients forming a principal ideal domain; over a general ring it becomes a spectral sequence.
The 2-torus is S^1 cross S^1. Since H_*(S^1) is Z in degrees 0 and 1, Künneth gives H_0(T^2) = Z, H_1(T^2) = Z tensor Z plus Z tensor Z = Z^2, H_2(T^2) = Z tensor Z = Z. More generally H_k(T^n) = Z^{(n choose k)}, so the Betti numbers of the n-torus are the binomial coefficients.
Künneth gives the n-torus Betti numbers as the binomial coefficients 'n choose k'.
Over a field there is no torsion term, but over the integers you must include the Tor correction whenever a factor has torsion — omitting it gives wrong answers. The splitting is non-natural, and the clean form needs the coefficients to be a principal ideal domain.