Based cellular chains of a universal cover as finite free right group-ring modules

Let $X$ be a nonempty connected finite CW complex with supplied characteristic maps, let $x\in X$ be a basepoint, put $\pi ={\pi }_1(X,x)$, and let $p:\tilde{X}⟶X$ be a chosen universal cover, which exists for the spaces considered here because a finite CW complex is locally path-connected and semilocally simply connected (Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover). Write $R=\mathbb{Z}[\pi ]$ for the integral group ring, a unital ring with basis the classes $[g]$ of the group elements (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements, The group ring $R[G]$ is a unital $R$-algebra with basis $G$, and each $g\in G$ is a unit of $R[G]$).

Lifted CW structure. Every open cell $e\subseteq X$ is contractible, so $p^{-1}(e)$ splits into components each mapped homeomorphically onto $e$; such a component is an open cell of $\tilde{X}$ over $e$. To obtain its lifted characteristic map, choose a point above the image of one interior point of the characteristic disk and lift the entire characteristic map $D^n\to X$ through $p$; this is possible because $D^n$ is simply connected (Lifting criterion for maps from path-connected locally path-connected spaces), and its interior maps homeomorphically onto the chosen component over $e$. The inverse $e\to p^{-1}(e)$ alone cannot be composed with the characteristic map on its boundary, where that inverse is undefined. These lifted characteristic maps give the standard lifted CW structure (Cell attachment by a characteristic map, Skeleta, CW subcomplexes, and relative CW complexes). Its $n$-skeleton is ${\tilde{X}}^n:=p^{-1}(X^n),$ the union of the closed lifted cells over the cells of $X$ of dimension at most $n$; for a CW pair $(X,A)$ the preimage $p^{-1}(A)$ is a CW subcomplex of $\tilde{X}$ and $p^{-1}(X^n\cup A)=p^{-1}(X^n)\cup p^{-1}(A)$ is the $n$-skeleton of the relative lifted structure.

Deck action. Identify $\pi$ with the deck group of $p$ by the no-reversal isomorphism of For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group, writing $T_g$ for the covering homeomorphism attached to $g\in \pi$; thus $T_1=\mathrm{i}\mathrm{d}$, $T_{gh}=T_g\circ T_h$, and every $T_g$ carries lifted cells onto lifted cells of the same dimension. A deck transformation is determined by its value at one point and acts freely, so the lifts of a single cell $e$ are exactly the cells $T_g\tilde{e}$ for one chosen lift $\tilde{e}$.

Right group-ring action. The action of the deck group on the singular and cellular chains of $\tilde{X}$ is written on the left and the length-preserving insertion of inverses makes it a right action of $R$ by $c\cdot g:=T_g^{-1}(c)=T_{g^{-1}}(c),g\in \pi ,$ extended $\mathbb{Z}$-bilinearly in $c$ and the group-ring coefficient (Relative singular homology, Oriented cellular chain group); in particular $(c\cdot g)\cdot h=c\cdot gh$. Each $T_{g^{-1}}$ is a homeomorphism of pairs $({\tilde{X}}^n,{\tilde{X}}^{n-1})\to ({\tilde{X}}^n,{\tilde{X}}^{n-1})$ and of pairs $({\tilde{X}}^n\cup p^{-1}A,{\tilde{X}}^{n-1}\cup p^{-1}A)$, so the action passes to the homology of those pairs.

Based cellular chains. For each cell $e$ of $X$ choose an orientation of $e$, that is, an orientation of the disk of its characteristic map, and choose one oriented lift $\tilde{e}$ carrying that orientation. Put $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(\tilde{X};R):=H_n({\tilde{X}}^n,{\tilde{X}}^{n-1};\mathbb{Z}),$ $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(\tilde{X},p^{-1}(A);R):=H_n({\tilde{X}}^n\cup p^{-1}(A),{\tilde{X}}^{n-1}\cup p^{-1}(A);\mathbb{Z})\mathrm{.}$ The notation after the semicolon records the deck-induced $R$-module structure; the homology coefficients are integral. By Relative homology of consecutive CW skeleta these integral homology groups are free abelian on the lifted cells. The right action above makes them finite free right $R$-modules on one chosen oriented lift of each cell of $X$ (respectively each relative cell of $(X,A)$): the lifts of one cell form the $\pi$-orbit $\{T_g\tilde{e}\}$, and $[g]\mapsto T_{g^{-1}}\tilde{e}$ bijects $\pi$ with that orbit. Using $R$ as a second homology coefficient group here would duplicate the lift-indexed generators and would not yield the claimed rank. Degrees without relative cells give the zero module, and for a disconnected finite $X$ the construction is applied componentwise with its component group ring.