Künneth theorem

In mathematics, the Künneth theorem of algebraic topology describes the singular homology of the cartesian product X × Y of two topological spaces, in terms of singular homology groups H_i(X, R) and H_j(X, R). In this article, R is a commutative ring of coefficients, and will be suppressed from the notation in homology groups. Even in the case where R is the ring Z of integers, the statement of the full result requires some use of homological algebra, namely use of the Tor functors. From now on the coefficients in R will always be tacitly understood in the notation.

Case of a field

If R is taken to be a field then there is no need to invoke the Tor functors. The result in this case can be used as a 'first approximation' to the general case. It states that

$H_{k}(X\times Y)\cong \bigoplus _{i+j=k}H_{i}(X)\otimes H_{j}(Y)$

Further there is a cross product operation showing how 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). The statement of the Künneth theorem, when R is a field, is that this linear mapping is an isomorphism.

Betti numbers of a product

As a consequence the Betti numbers of X × Y are determined by those of X and of Y; the statement is formally equivalent to saying that if p_Z(t) is the generating function of the sequence of Betti numbers b_k(Z) of a space Z, then

$p_{X\times Y}(t)=p_{X}(t)p_{Y}(t)\,\!$.

Here when there are finitely many Betti numbers of X and Y, each of which is a natural number rather than ∞, this reads as an identity on Poincaré polynomials. In the general case these are formal power series with possible coefficients ∞, and have to be interpreted accordingly. Furthermore, the Betti numbers over any field F, b_k(Z,F), satisfy the same type of relationship as the usual b_k(Z,Q) for rational number coefficients (they need not actually be the same numbers unless the homology is torsion-free).

Formulation of general case

To extend this to the case of general R, it is necessary to change the statement: the R-module homomorphism defined φ in just the same way by the cross product is injective, and there is a description now of its cokernel. That is, we have to define an R-module

$T=\bigoplus _{p+q=k-1}\operatorname {Tor} _{R}(H_{p}(X),H_{q}(Y))$

Then the cokernel of φ is isomorphic to T.

Therefore, in any case where the relevant Tor groups can be shown to vanish, we do have an isomorphism. This is not, however, universally true. (In the early days of algebraic topology the phenomena caused by torsion in homology groups, of which this is one, appeared subtle and misled researchers.)

For homology, we have the following

Theorem (Künneth). If X and Y are CW complexes and R is a principal ideal domain, then there are natural short exact sequences

$0\rightarrow \bigoplus _{i}\left(H_{i}(X;R)\otimes _{R}H_{n-i}(Y;R)\right)\rightarrow H_{n}(X\times Y;R)\rightarrow \bigoplus _{i}\mathrm {Tor} _{R}\left(H_{i}(X;R),H_{n-i-1}(Y;R)\right)\rightarrow 0$

and these sequences split (but not canonically).

The usual proof of the result depends on the Eilenberg–Zilber theorem.

The result is named for the German mathematician Otto Hermann Künneth (1892–1975). The idea of a Künneth formula has now become a generic term, applied to many homological theories.