Künneth theorem

Künneth theorem

In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem 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 "X" × "Y", but Künneth theorems are true in many different homology and cohomology theories. Künneth theorems are named for the German mathematician Otto Hermann Künneth (1892–1975).

Singular homology with coefficients in a field

Let "X" and "Y" be two topological spaces, and let "F" be a field. In this situation, the Künneth theorem for singular homology states that for any integer "k",:igoplus_{i + j = k} H_i(X, F) otimes H_j(Y, F) cong H_k(X imes Y, F).Furthermore, the isomorphism is natural isomorphism. The map from the sum to the homology group of the product is called the cross product.

A consequence of this 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 "p""Z"("t") is the generating function of the sequence of Betti numbers "b""k"("Z") of a space "Z", then

: p_{X imes 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 ∞, 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 be different.)

Singular homology with coefficients in a PID

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 are a PID.

In this case the equation above is no longer true. Instead a correction factor appears to account for the possibility of torsion phenomena. For example, if H_1(X, mathbf{Z}) = mathbf{Z}/(2) and H_1(Y, mathbf{Z}) = mathbf{Z}/(3), then the tensor product of these homology groups will be zero. But the second homology of "X" × "Y" will "always" contain a correction factor to account for the vanishing of this product. This correction factor is expressed in terms of the Tor functor, the first derived functor of the tensor product.

When "X" and "Y" are CW complexes and "R" is a PID, then the correct statement of the Künneth theorem is that there are natural short exact sequences:0 arr igoplus_{i + j = k} H_i(X,R) otimes_R H_j(Y, R) arr H_k(X imes Y,R) arr igoplus_{i + j = k-1} mathrm{Tor}_1^R(H_i(X, R), H_j(Y, R)) arr 0.Furthermore these sequences split, but not canonically.

The Künneth spectral sequence

For general "R", the homology of "X" and "Y" is related to the homology of their product by a spectral sequence. In the cases described above, this spectral sequence collapses to give an isomorphism or a short exact sequence. The Künneth spectral sequence is:E_{pq}^2 = igoplus_{q_1 + q_2 = q} mathrm{Tor}^R_p(H_{q_1}(X, R), H_{q_2}(X, R)) Rightarrow H_{p+q}(X imes Y, R).

The Künneth formula in the derived category

A much cleaner statement of the Künneth formula becomes possible in the derived category. In this case, the formula becomes a natural isomorphism between quasi-isomorphism classes of singular chain complexes::C_*(X) otimes^{mathbf L} C_*(Y) cong C_*(X imes Y).Here otimes^{mathbf L} denotes the derived tensor product.

Künneth theorems in other homology and cohomology theories

The above statements are also true for singular cohomology and sheaf cohomology. For sheaf cohomology on an algebraic variety, Grothendieck found six spectral sequences relating the possible hyperhomology groups of two chain complexes of sheaves and the hyperhomology groups of their tensor product. (See EGA III2, Théorème 6.7.3) Künneth theorems are also true for K-theory, cobordism, and l-adic cohomology.

References

*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1963
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Seconde partie
journal = Publications Mathématiques de l'IHÉS
volume = 17
pages = 5–91
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1963__17_

External links

*Springer|title=Künneth formula|id=k/k056010


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