Editing Künneth theorem (section)

== Singular homology with coefficients in a field ==
Let ''X'' and ''Y'' be two topological spaces.  In general one uses singular homology; but if ''X'' and ''Y'' happen to be [[CW complex]]es, then this can be replaced by [[cellular homology]], because that is isomorphic to singular homology. The simplest case is when the coefficient ring  for homology is a field ''F''. In this situation, the Künneth theorem (for singular homology) states that for any integer ''k'',

:<math>\bigoplus_{i + j = k} H_i(X; F) \otimes H_j(Y; F) \cong H_k(X \times Y; F)</math>.

Furthermore, the isomorphism is a [[natural isomorphism]]. The map from the sum to the homology group of the product is called the ''cross product''. More precisely, there is a cross product operation by which an ''i''-cycle on ''X'' and a ''j''-cycle on ''Y'' can be combined to create an  <math>(i+j)</math>-cycle on <math>X \times Y</math>; so that there is an explicit linear mapping defined from the direct sum to <math>H_k(X \times Y)</math>.

A consequence of this result is that the [[Betti number]]s, the dimensions of the homology with <math>\Q</math> coefficients, of <math>X \times Y</math> can be determined from those of ''X'' and ''Y''. If <math>p_Z(t)</math> is the [[generating function]] of the sequence of Betti numbers <math>b_k(Z)</math> of a space ''Z'', then

: <math>p_{X \times Y}(t) = p_X(t) p_Y(t).</math>

Here when there are finitely many Betti numbers of ''X'' and ''Y'', each of which is a [[natural number]] rather than <math>\infty</math>, this reads as an identity on [[Poincaré polynomial]]s. In the general case these are [[formal power series]] with possibly infinite coefficients, and have to be interpreted accordingly. Furthermore, the above statement holds not only for the Betti numbers but also for the generating functions of the dimensions of the homology over any field. (If the integer homology is not [[torsion (algebra)|torsion-free]], then these numbers may differ from the standard Betti numbers.)