Product homology is tensor product plus a one-degree-shifted Tor correction.Suggest a slogan
Let \(X, Y\) be CW complexes and \(R\) a principal ideal domain. For each \(k\) there is a natural short exact sequence \[ 0 \to \bigoplus_{i+j=k} H_{i}(X; R) \tensor_{R} H_{j}(Y; R) \to H_{k}(X\cross Y; R) \to \bigoplus_{i+j=k-1} \Tor^R_1\qty{H_{i}(X; R), H_{j}(Y; R)} \to 0 ,\] and it splits, though not naturally: \[ H_{k}(X\cross Y; R) \cong \qty{ \bigoplus_{i+j = k} H_{i}(X;R) \tensor_R H_{j}(Y;R) } \oplus \bigoplus_{i+j = k-1}\Tor^R_1\qty{H_{i}(X;R), H_{j}(Y;R)} \] [@Hat02].