# A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces.

The Künneth-Theorem which I know states: If $R$ is a PID, for any topological space $X$ and $Y$ there is a shot exact sequence $$0 \to \bigoplus_{i + j = k} H_i(X; R) \otimes_R H_j(Y; R) \to H_k(X \times Y; R) \to \bigoplus_{i + j = k-1} \mathrm{Tor}_1^R(H_i(X; R), H_j(Y; R)) \to 0.$$

The sequence splits and therefore we have a formula for product spaces in singular homology over a PID $R$.

My question is, is there a version in singular cohomology for pairs of product spaces to determine something like $H^*(X\times Y,A\times Y;R)$, where $X,Y$ are topological spaces, $A\subset X$ is a subspace and $R$ is a PID? For references I would be happy.

One of the starting points of my investigation is a similar question which appears on MSE today https://math.stackexchange.com/questions/1642869/hx-ar-cong-hx-ar-rightarrow-hx-times-y-a-times-yr-cong-h (and some other problems).

Regards.

• It is true, and is proven in the same way with notational changes: Corollary 12.10 of Dold's Lectures on Algebraic Topology. – Chris Gerig Feb 7 '16 at 0:24
• thanks. I will try to have look in this book as soon as possible. – Sabrina Gemsa Feb 7 '16 at 0:45
• You should be careful. The cohomology version typically also requires some sort of finite-generation condition on the cohomology. I agree Dold is a good reference, but I think it should be to Proposition VII.7.6 for what you want. Chris's reference is, I think, missing a Roman numeral chapter number. – Greg Friedman Feb 7 '16 at 4:48
• @Greg Friedman thank you. I will be careful. – Sabrina Gemsa Feb 7 '16 at 11:35
• You can also consult Spanier, page 249. – Bruno Stonek Feb 7 '16 at 12:28