Universal-cover boundaries, maps and homotopies respect the right group-ring action

Let $z\in C_n(\tilde{X})=H_n({\tilde{X}}^n,{\tilde{X}}^{n-1})$ be represented by a relative cycle $c\in S_n({\tilde{X}}^n)$ with $\mathrm{\partial }c\in S_{n-1}({\tilde{X}}^{n-1})$, and let $g\in \pi$. By [F4] the connecting homomorphism satisfies $\delta [c]=[\mathrm{\partial }c]$ and is natural for the map of pairs $T_{g^{-1}}:({\tilde{X}}^n,{\tilde{X}}^{n-1})\to ({\tilde{X}}^n,{\tilde{X}}^{n-1})$, so $\delta (T_{g^{-1}}{)}_{\ast }[c]=(T_{g^{-1}}{)}_{\ast }\delta [c]$, using [F3]; the quotient map $H_{n-1}({\tilde{X}}^{n-1})\to H_{n-1}({\tilde{X}}^{n-1},{\tilde{X}}^{n-2})$ is likewise natural because it is induced by the inclusion of chain complexes. Hence with [F1] and [F2], $d_n(c\cdot g)=d_n(c)\cdot g$.

Let $H:X\times I\to Y$ be a cellular homotopy from $f$ to $g$, and let $\tilde{H}:\tilde{X}\times I\to \tilde{Y}$ be the lift of $H\circ (p\times \mathrm{i}\mathrm{d})$ with $\tilde{H}(-,0)=\tilde{f}$, which exists and is unique by [F6] since $\tilde{X}\times I$ is connected. Its end $\tilde{H}(-,1)$ is a lift of $g\circ p$, so it equals $T_{\beta }^{\mathrm{\prime }}\circ \tilde{g}$ for a unique $\beta \in {\pi }^{\mathrm{\prime }}$ by [F6] and the free transitive deck action.

Applying step 1.1 with $g=h^{-1}$ gives $d_n(T_{h}c)=T_h{d}_n(c)$. Thus $T_h$ is an automorphism of the underlying integral cellular chain complex, with inverse $T_{h^{-1}}$. The right-action convention gives $T_h(c\cdot g)=T_h{T}_{g^{-1}}c=T_{h{g}^{-1}}c=T_{(hg{h}^{-1}{)}^{-1}}{T}_{h}c=T_h(c)\cdot (hg{h}^{-1})$. This is semilinearity, not right-linearity for the fixed coefficients; the selected finite right-module basis can also change under $T_h$.

Let $\tilde{f}:\tilde{X}\to \tilde{Y}$ be a lift of the cellular $f$. Since $f$ is cellular, $\tilde{f}({\tilde{X}}^n)\subseteq {\tilde{Y}}^n$ for every $n$: because $f(X^n)\subseteq Y^n$ and $q\tilde{f}=fp$; an individual cell may map across several target cells or collapse. Hence ${\tilde{f}}_{\mathrm{\#}}$ carries $S_n({\tilde{X}}^n)$ into $S_n({\tilde{Y}}^n)$ and $S_n({\tilde{X}}^{n-1})$ into $S_n({\tilde{Y}}^{n-1})$, so it induces maps $C_n(\tilde{f}):H_n({\tilde{X}}^n,{\tilde{X}}^{n-1})\to H_n({\tilde{Y}}^n,{\tilde{Y}}^{n-1})$ on relative homology by naturality of the connecting homomorphisms and quotient maps as in step 1.1.

For $\sigma \in S_n({\tilde{X}}^m)$ with $m\le n$ one has $\tilde{H}(\sigma \times I)\subseteq {\tilde{Y}}^{m+1}$, because $H$ is cellular and $X^m\times I\subseteq (X\times I{)}^{m+1}$; hence the prism operator $P_{\tilde{H}}$ of [F5] maps $S_n({\tilde{X}}^m)$ into $S_{n+1}({\tilde{Y}}^{m+1})$. In particular $P_{\tilde{H}}$ maps $S_n({\tilde{X}}^n)$ into $S_{n+1}({\tilde{Y}}^{n+1})$ and $S_n({\tilde{X}}^{n-1})$ into $S_{n+1}({\tilde{Y}}^n)$, so it induces $s_n:C_n(\tilde{X})\to C_{n+1}(\tilde{Y})$ by $s_n[z]:=[P_{\tilde{H}}z]$.

The induced maps of step 2.2 commute with the differentials: for a relative cycle $c$ as in step 1.1, $\mathrm{\partial }{\tilde{f}}_{\mathrm{\#}}c={\tilde{f}}_{\mathrm{\#}}\mathrm{\partial }c$ by [F3], and naturality of $\delta$ and of the quotient map gives $d_n^{\tilde{Y}}{C}_n(\tilde{f})[c]=C_{n-1}(\tilde{f})d_n^{\tilde{X}}[c]$.

For $g\in \pi$ the map $h\mapsto f_{\ast }(h)$ is a group isomorphism and $\tilde{f}\circ T_g=T_{f_{\ast }(g)}^{\mathrm{\prime }}\circ \tilde{f}$ by [F7], so on chains $C_n(\tilde{f})(c\cdot g)={\tilde{f}}_{\mathrm{\#}}{T}_{g^{-1}}c=T_{f_{\ast }(g{)}^{-1}}^{\mathrm{\prime }}{\tilde{f}}_{\mathrm{\#}}c=C_n(\tilde{f})(c)\cdot f_{\ast }(g)$; thus $C_{\ast }(\tilde{f})$ is right $\mathbb{Z}[{\pi }^{\mathrm{\prime }}]$-linear for the transported source structure.

The class in step 2.3 is well defined: if $z$ is replaced by $z+\mathrm{\partial }w$ with $w\in S_{n+1}({\tilde{X}}^n)$, then by [F5] $P_{\tilde{H}}\mathrm{\partial }w=(T_{\beta }^{\mathrm{\prime }}\circ \tilde{g}{)}_{\mathrm{\#}}w-{\tilde{f}}_{\mathrm{\#}}w-\mathrm{\partial }{P}_{\tilde{H}}w$, where the first two terms lie in $S_{n+1}({\tilde{Y}}^n)$ (step 2.2) and the last is a boundary in the relative complex $({\tilde{Y}}^{n+1},{\tilde{Y}}^n)$; adding a chain of $S_n({\tilde{X}}^{n-1})$ changes $P_{\tilde{H}}z$ by an element of $S_{n+1}({\tilde{Y}}^n)$.

Applying the identity of [F5] to $z$ and reducing modulo the subcomplexes defining the relative groups gives $d_{n+1}^{\tilde{Y}}{s}_n[z]+s_{n-1}{d}_n^{\tilde{X}}[z]=(T_{\beta }^{\mathrm{\prime }}\circ \tilde{g}{)}_{\ast }[z]-{\tilde{f}}_{\ast }[z]$ in $C_n(\tilde{Y})$, which is the displayed chain-homotopy identity; equivalently $C_{\ast }(\tilde{f})\simeq T_{\beta }^{\mathrm{\prime }}\circ C_{\ast }(\tilde{g})$.

The operators of step 2.3 are right-linear: $\tilde{H}\circ (T_h\times \mathrm{i}\mathrm{d})=T_{f_{\ast }(h)}^{\mathrm{\prime }}\circ \tilde{H}$ for $h\in \pi$ by [F6] and [F7], since both sides are lifts agreeing at time zero, so $s_n(c\cdot h)=s_n(c)\cdot f_{\ast }(h)$ exactly as in step 3.2. At time one this also proves that the composite $T_{\beta }^{\mathrm{\prime }}{C}_{\ast }(\tilde{g})$ is right-linear for $f_{\ast }$; it does not assert that $T_{\beta }^{\mathrm{\prime }}$ is right-linear for the unmodified $g_{\ast }$-module. If $H$ fixes $x$ and both endpoint lifts send $\tilde{x}$ to $\tilde{y}$, uniqueness at $\tilde{x}$ gives $\beta =1$.

Clause 1 is steps 1.1 and 2.1, clause 2 is steps 2.2, 3.1 and 3.2, and clause 3 is steps 1.2, 2.3, 3.3, 4.1 and 4.2; every step used only finite CW approximation [F8] where cellular maps were assumed, and no step used a choice principle.