Let \(X\) be a CW complex with \(n\)-skeleta \(X^n\), and \(X^{-1}\coloneqq\emptyset\). The cellular chain complex of \(X\) has \(C_n^{\cell}(X) \coloneqq H_n(X^n, X^{n-1})\) and differential \(d_n\colon C_n^{\cell}(X)\to C_{n-1}^{\cell}(X)\) the composite \[ H_n(X^n, X^{n-1}) \mapsvia{\del_n} H_{n-1}(X^{n-1}) \mapsvia{j_{n-1}} H_{n-1}(X^{n-1}, X^{n-2}) \] of the boundary map of the long exact sequence of the pair \((X^n, X^{n-1})\) and the map induced by inclusion. The cellular homology \(H_n^{\cell}(X)\) is the homology of \((C_*^{\cell}(X), d_*)\).
The group \(C_n^{\cell}(X)\) is free abelian with a basis in bijection with the \(n\)-cells of \(X\), and \(d_{n-1}\circ d_n = 0\). By the cellular boundary formula, for \(n\geq 2\) the coefficient of the \((n-1)\)-cell \(e^{n-1}_\beta\) in \(d_n(e^n_\alpha)\) is the degree of the composite \(S^{n-1}_\alpha \to X^{n-1}\to X^{n-1}/\qty{X^{n-1}\sm e^{n-1}_\beta} \cong S^{n-1}_\beta\) of the attaching map of \(e^n_\alpha\) and the quotient map. There is an isomorphism \(H_n^{\cell}(X)\cong H_n(X)\) for every \(n\) [@Hat02].
[@Hat02].