Kunneth theorem

Let X, Y be topological spacesMathworldPlanetmath. One can ask a question: how homologyMathworldPlanetmathPlanetmath of X×Y are related to homologies of X and Y. The answer to this question depends on the homology theory we’re talking about and also the coefficients ring. On the other hand, it is well known, that all homology theories are isomorphic on CW-complexesMathworldPlanetmath. Thus we may restrict to CW-complexes. Nevertheless the following theorem is more general:

Theorem. (Kunneth) Assume, that X, Y are topological spaces and R is a principal ideal domainMathworldPlanetmath. Denote by H*(X,R) the singular homology with coefficients in R. Then, for any k>0 there exists following short exact sequenceMathworldPlanetmathPlanetmath in the categoryMathworldPlanetmath of R-modules:


where Tor denotes the Tor functor. Furthermore this sequence splits, i.e. the middle term is a direct sumPlanetmathPlanetmath (up to an isomorphismMathworldPlanetmath) of left and right term.

It should be mentioned, that if R=𝔽 is a field, then the Tor functor is always trivial (i.e. Tor1𝔽(M,N)=0 for all vector spaces M,N over 𝔽) and in this case Kunneth formula can be stated as


for any k>0.

Title Kunneth theorem
Canonical name KunnethTheorem
Date of creation 2013-03-22 19:13:57
Last modified on 2013-03-22 19:13:57
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type Theorem
Classification msc 55N10
Synonym Kunneth formula
Synonym Kunneth’s formula
Synonym Kunneth’s theorem
Synonym Kunneth theorem