Cellular chains of the maximal abelian cover compute the link Alexander module

A finite CW structure has finitely many open cells. Each open cell is contractible, so its inverse image under a covering is a disjoint union of copies of that cell. The regular deck group $H$ acts freely and transitively on those copies. Choose one lifted copy for each cell; finitely many choices suffice. Every other lift is uniquely a deck translate of the chosen lift, so the lifted cellular chain group in degree $n$ is a free $\mathbb{Z}[H]={\mathrm{\Lambda }}_r$-module with one basis element per $n$-cell of $E_L$.

Translating a lifted characteristic map by $h\in H$ translates all of its oriented boundary cells by the same $h$. Thus every cellular boundary map is ${\mathrm{\Lambda }}_r$-linear. The cellular identity ${\mathrm{\partial }}^2=0$ then makes these finite-rank free modules a chain complex over ${\mathrm{\Lambda }}_r$.

Each deck transformation is cellular on the lifted CW structure. By the naturality in [F3], the cellular-to-singular homology isomorphism commutes with every deck transformation and hence is ${\mathrm{\Lambda }}_r$-linear. In degree one it identifies the homology of the lifted cellular complex with $H_1(E_L^{\mathrm{a}\mathrm{b}};\mathbb{Z})$.

Under the Axiom-of-Choice convention [A1], [F2] says inclusion of the restricted maximal abelian cover into $X_L^{\mathrm{a}\mathrm{b}}$ induces a ${\mathrm{\Lambda }}_r$-module isomorphism on first homology. Therefore the lifted cellular complex has first homology $A_L$ with the stated deck action.