The Künneth Formula

Abstract

The Künneth formula has its historic origin in algebraic topology. Given two topological spaces X and Y, we may ask how the (singular) homology groups of their topological product X × Y is related to the homology groups of X and Y. This question may be answered by separating the problem into two parts. If C(X), C(Y), C(X × Y) stand for the singular chain complexes of X, Y, X × Y respectively, then a theorem due to Eilenberg-Zilber establishes that the chain complex C(X × Y) is canonically homotopy-equivalent to the tensor product of the chain complexes C(X) and C(Y),

$$ C(X \times Y) \simeq C(X) \otimes C(Y) $$

(foC(X) and C(Y) to the homology groups of C(X) and C(Y). This relation is furnished by the Künneth formula, whose validity we establish under much more general circumstances than would be required by the topological situation. For, in that case, we are concerned with free chain complexes of ℤ-modules; the argument we give permits arbitrary chain complexes C, D of Λ-modules, where Λ is any p.i.d., provided only that one of C, D is flat.