Cellular chains of $X\times Y$ are the graded tensor product of cellular chains, with boundary $d(e^i\times e^j) = de^i\times e^j + (-1)^i e^i\times de^j$.Suggest a slogan
Let \(X\) and \(Y\) be CW complexes such that \(X\cross Y\) with the product topology is a CW complex with cells \(e^i_\alpha\cross e^j_\beta\); this holds if \(X\) or \(Y\) is locally compact, or if both have countably many cells [@Hat02]. Then the cellular chain complexes satisfy \[ C_n(X \cross Y) \cong \bigoplus_{i+j=n} C_i(X) \tensor_\ZZ C_j(Y), \qquad e^i\cross e^j\mapsto e^i\tensor e^j ,\] with boundary \[ d(e^i\cross e^j) = de^i\cross e^j + (-1)^{i} e^i\cross de^j \] [@Hat02]. In particular, if \(X\) and \(Y\) have finitely many cells in each dimension and \(p_X(t) = \sum_n a_n t^n\), \(p_Y(t)\) count their \(n\)-cells, then \(p_{X\cross Y}(t) = p_X(t)\,p_Y(t)\).