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Concept
Künneth theorem
Summary
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 . 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.
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,
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 -cycle on ; 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 coefficients, of can be determined from those of X and Y. If is the generating function of the sequence of Betti numbers of a space Z, then
Here when there are finitely many Betti numbers of X and Y, each of which is a natural number rather than , 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.
Official source
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