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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''. 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'', : 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'')-cycle on ''X'' × ''Y''; so that there is an explicit linear mapping defined from the direct sum to ''H''''k''(''X'' × ''Y''). A consequence of this result is that the Betti numbers, the dimensions of the homology with Q coefficients, of ''X'' × ''Y'' can be determined from those of ''X'' and ''Y''. If ''pZ''(''t'') is the generating function of the sequence of Betti numbers ''bk''(''Z'') 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. 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.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Künneth theorem」の詳細全文を読む スポンサード リンク
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