Cellular chain group ($C_q^{\mathrm{cell}}(X)=H_q(X^q,X^{q-1};\mathbb{Z})$)

For a CW complex, the qth cellular chain group is the relative homology group of successive skeleta, a free abelian group with one generator for each q-cell. The attaching maps determine the cellular boundary, and the resulting cellular chain complex computes singular homology.