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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 http://math.stackexchange.com/questions/1642869/hx-ar-cong-hx-ar-rightarrow-hx-times-y-a-times-yr-cong-hhttps://math.stackexchange.com/questions/1642869/hx-ar-cong-hx-ar-rightarrow-hx-times-y-a-times-yr-cong-h (and some other problems).

Regards.

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 http://math.stackexchange.com/questions/1642869/hx-ar-cong-hx-ar-rightarrow-hx-times-y-a-times-yr-cong-h (and some other problems).

Regards.

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.

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Sabrina Gemsa
Sabrina Gemsa
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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 http://math.stackexchange.com/questions/1642869/hx-ar-cong-hx-ar-rightarrow-hx-times-y-a-times-yr-cong-h (and some other problems).

Regards.