In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer.
These many results are named for the German mathematician Hermann Künneth.
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, Furthermore, the isomorphism is a natural isomorphism.
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
; so that there is an explicit linear mapping defined from the direct sum to
A consequence of this result is that the Betti numbers, the dimensions of the homology with
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.
The above formula is simple because vector spaces over a field have very restricted behavior.
A correction factor appears to account for the possibility of torsion phenomena.
The short exact sequences just described can easily be used to compute the homology groups with integer coefficients of the product
for brevity's sake, one knows from a simple calculation with cellular homology that The only non-zero Tor group (torsion product) which can be formed from these values of
For cellular chains on CW complexes, it is a straightforward isomorphism.
[1] The freeness of the chain modules means that in this geometric case it is not necessary to use any hyperhomology or total derived tensor product.
For sheaf cohomology on an algebraic variety, Alexander Grothendieck found six spectral sequences relating the possible hyperhomology groups of two chain complexes of sheaves and the hyperhomology groups of their tensor product.
[2] There are many generalized (or "extraordinary") homology and cohomology theories for topological spaces.
Unlike ordinary homology and cohomology, they typically cannot be defined using chain complexes.
Thus Künneth theorems can not be obtained by the above methods of homological algebra.
Nevertheless, Künneth theorems in just the same form have been proved in very many cases by various other methods.
The first were Michael Atiyah's Künneth theorem for complex K-theory and Pierre Conner and Edwin E. Floyd's result in cobordism.
[3][4] A general method of proof emerged, based upon a homotopical theory of modules over highly structured ring spectra.