In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y and their product space X × Y {\displaystyle X\times Y} . 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. A Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the name has become generic. These many results are named for the German mathematician Hermann Künneth.
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 complexes, 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,
⨁ i + j = k H i ( X ; F ) ⊗ H j ( Y ; F ) ≅ H k ( X × Y ; F ) {\displaystyle \bigoplus _{i+j=k}H_{i}(X;F)\otimes H_{j}(Y;F)\cong H_{k}(X\times Y;F)} . 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 ( i + j ) {\displaystyle (i+j)} -cycle on X × Y {\displaystyle X\times Y} ; so that there is an explicit linear mapping defined from the direct sum to H k ( X × Y ) {\displaystyle H_{k}(X\times Y)} . A consequence of this result is that the Betti numbers, the dimensions of the homology with Q {\displaystyle \mathbb {Q} } coefficients, of X × Y {\displaystyle X\times Y} can be determined from those of X and Y. If p Z ( t ) {\displaystyle p_{Z}(t)} is the generating function of the sequence of Betti numbers b k ( Z ) {\displaystyle b_{k}(Z)} of a space Z, then
p X × Y ( t ) = p X ( t ) p Y ( t ) . {\displaystyle p_{X\times Y}(t)=p_{X}(t)p_{Y}(t).}
Here when there are finitely many Betti numbers of X and Y, each of which is a natural number rather than ∞ {\displaystyle \infty } , this reads as an identity on Poincaré polynomials. 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-free, then these numbers may differ from the standard Betti numbers.)
Singular homology with coefficients in a principal ideal domain The above formula is simple because vector spaces over a field have very restricted behavior. As the coefficient ring becomes more general, the relationship becomes more complicated. The next simplest case is the case when the coefficient ring is a principal ideal domain. This case is particularly important because the integers Z {\displaystyle \mathbb {Z} } form a PID. In this case the equation above is no longer always true. A correction factor appears to account for the possibility of torsion phenomena. This correction factor is expressed in terms of the Tor functor, the first derived functor of the tensor product. When R is a PID, then the correct statement of the Künneth theorem is that for any topological spaces X and Y there are natural short exact sequences
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