Two high relative cell layers have free homotopy bases and their cellular boundary matrix

The situation of the statement is the one produced by [F1]: for a finite relative homotopy equivalence the cell-trading lemma yields, relative to $L$, a formal deformation to a pair whose relative cells lie in two adjacent degrees, with the deformation respecting the homotopy class of the inclusion and transporting the relative torsion, and this statement therefore applies to each pair so produced; conversely it applies verbatim to any pair satisfying its two-layer hypothesis. All claims below concern a pair $(K,L)$ with relative cells only in the two degrees $n,n+1\ge 3$, a vertex basepoint $k_0\in L$, and $\pi ={\pi }_1(K,k_0)$.

Let $p:\tilde{K}\to K$ be a universal cover and put $\tilde{L}=p^{-1}(L)$ and ${\tilde{K}}_n=p^{-1}(K_n)$, where $K_n=L\cup K^{(n)}=L\cup \{\text{relative }n\text{-cells}\}$. Then $\tilde{L}$ and ${\tilde{K}}_n$ are CW subcomplexes of the lifted CW structure, ${\tilde{K}}_n=\tilde{L}\cup {\tilde{K}}^{(n)}$ is carried by $\tilde{L}$ and the lifts of the relative $n$-cells, and $\tilde{K}$ is carried by ${\tilde{K}}_n$ and the lifts of the relative $(n+1)$-cells. This distinction matters when $L$ has cells above dimension $n$. The deck group is identified with $\pi$ and acts freely on $\tilde{K}$, the lifts of a single cell forming one orbit $\{T_g\tilde{e}\}$, and the right action on chains is $c\cdot g=T_{g^{-1}}(c)$, which passes to the homology of the pairs $({\tilde{X}}^m\cup p^{-1}A,{\tilde{X}}^{m-1}\cup p^{-1}A)$.

The restrictions $p:\tilde{L}\to L$ and $p:{\tilde{K}}_n\to K_n$ are coverings: if $U$ is an evenly covered neighbourhood of a point $x\in K_n$ with sheets $V$ over $U$, then $p^{-1}(U\cap K_n)$ is the disjoint union of the sets $V\cap {\tilde{K}}_n$, each mapped homeomorphically onto $U\cap K_n$ because $p(V\cap {\tilde{K}}_n)=p(V)\cap p({\tilde{K}}_n)=U\cap K_n$.

Every relative $n$-cell of $K$ has attaching map with image in $K^{(n-1)}=L^{(n-1)}\subseteq L$, and $S^{n-1}$ is simply connected because $n\ge 3$, so the attaching map lifts to $\tilde{L}$ by the lifting criterion; hence the relative cells of $({\tilde{K}}_n,\tilde{L})$ are exactly the lifts of the relative $n$-cells, attached directly to $\tilde{L}$, and the relative cells of $(\tilde{K},{\tilde{K}}_n)$ are the lifts of the relative $(n+1)$-cells, attached directly to ${\tilde{K}}_n=\tilde{L}\cup {\tilde{K}}^{(n)}$ because $S^n$ is simply connected for $n\ge 3$.

The space $\tilde{L}$ is path-connected and simply connected. The monodromy action of ${\pi }_1(L,k_0)$ on the fibre $p^{-1}(k_0)$ is transitive, because the monodromy of a loop class $[\gamma ]$ is the deck transformation $T_{j_{\ast }[\gamma ]}$ of the universal cover [F15], the deck group is transitive on the lifts of the vertex $k_0$ [F2], and $j_{\ast }$ is onto since it is an isomorphism; hence the whole fibre over $k_0$ lies in one path component of $\tilde{L}$ [F17], and every point of $\tilde{L}$ is joined to that fibre by a lifted path [F16], so $\tilde{L}$ is path-connected. Also ${\pi }_1(\tilde{L},{\tilde{k}}_0)$ is trivial: for a loop $\tilde{\gamma }$ at ${\tilde{k}}_0$ with $\gamma =p\tilde{\gamma }$ one has $T_{j_{\ast }[\gamma ]}({\tilde{k}}_0)=\tilde{\gamma }(1)={\tilde{k}}_0$, so $j_{\ast }[\gamma ]=1$ and then $[\gamma ]=1$ because $j_{\ast }$ is injective, and a null-homotopy of $\gamma$ in $L$ lifts through the covering $\tilde{L}\to L$ to a null-homotopy of $\tilde{\gamma }$ because ${\pi }_1(D^2)$ is trivial [F6].

Applying the high-relative-cells lemma to the pair $({\tilde{K}}_n,\tilde{L})$, whose relative cells all have dimension $n\ge 3$, gives a bijection ${\pi }_0(\tilde{L})\to {\pi }_0({\tilde{K}}_n)$ and an isomorphism ${\pi }_1(\tilde{L},{\tilde{k}}_0)\to {\pi }_1({\tilde{K}}_n,{\tilde{k}}_0)$; by step 1.5 the space ${\tilde{K}}_n$ is therefore nonempty, path-connected and simply connected.

By the single-cell-layer lemma applied to $A=\tilde{L}$, $k=n$ and the lifted $n$-cells with their lifted characteristic maps, ${\pi }_n({\tilde{K}}_n,\tilde{L})$ and $H_n({\tilde{K}}_n,\tilde{L};\mathbb{Z})$ are free abelian on the lifts of the relative $n$-cells, with basis classes $c_{\tilde{e}}$ and $u_{\tilde{e}}$ satisfying $h(c_{\tilde{e}})=u_{\tilde{e}}=({\chi }_{\tilde{e}}{)}_{\ast }[D^n,S^{n-1}]$; the construction is choice-free and the class $c_{\tilde{e}}$ is independent of the choices made in representing it.

For $i\ge 2$ the covering $p:{\tilde{K}}_n\to K_n$ induces an isomorphism ${\pi }_i({\tilde{K}}_n,\tilde{x})\to {\pi }_i(K_n,p(\tilde{x}))$: it is injective because a null-homotopy of the composite of a based map $S^i\to {\tilde{K}}_n$ with $p$ lifts to a null-homotopy of that map by homotopy lifting [F8], and it is surjective because $S^i$ is simply connected for $i\ge 2$, so every based map $S^i\to K_n$ lifts through the covering by the lifting criterion [F6, F7]. The same argument applies to the coverings $\tilde{L}\to L$ and $\tilde{K}\to K$.

Suppose now that the inclusion $i:L\hookrightarrow K$ is a homotopy equivalence. By [F19] $K$ strongly deformation retracts onto $L$, and the time-one map $D_1:K\to L$ of that retraction satisfies $D_1\circ i={\mathrm{i}\mathrm{d}}_L$ and $i\circ D_1\simeq {\mathrm{i}\mathrm{d}}_K$, so by [F20] the map $i_{\ast }:{\pi }_r(L,k_0)\to {\pi }_r(K,k_0)$ is an isomorphism for every $r\ge 1$. Exactness of the pair sequences [F9] then gives ${\pi }_r(K,L)=0$ for every $r\ge 1$: for $r\ge 2$ both ${\pi }_r(L)\to {\pi }_r(K)$ and ${\pi }_{r-1}(L)\to {\pi }_{r-1}(K)$ are isomorphisms, so the kernel of ${\pi }_r(K,L)\to {\pi }_{r-1}(L)$ and the image of ${\pi }_r(K)\to {\pi }_r(K,L)$ vanish, while for $r=1$ the isomorphism ${\pi }_1(L)\to {\pi }_1(K)$ makes the pointed set ${\pi }_1(K,L)$ trivial since $L$ and $K$ are connected. In particular ${\pi }_n(K,L)=0$ and ${\pi }_{n+1}(K,L)=0$.

By the same lemma applied to $A={\tilde{K}}_n$, $k=n+1\ge 4$ and the lifted $(n+1)$-cells, ${\pi }_{n+1}(\tilde{K},{\tilde{K}}_n)$ and $H_{n+1}(\tilde{K},{\tilde{K}}_n;\mathbb{Z})$ are free abelian on the lifts of the relative $(n+1)$-cells, with $h(c_{\tilde{e}})=u_{\tilde{e}}=({\chi }_{\tilde{e}}{)}_{\ast }[D^{n+1},S^n]$.

The covering projection is a based map of pairs, so by functoriality of boundaries [F10] it induces a morphism between the long exact sequences of $({\tilde{K}}_n,\tilde{L})$ and $(K_n,L)$ and between those of $(\tilde{K},{\tilde{K}}_n)$ and $(K,K_n)$; these sequences are exact [F9], and the comparison maps in the degrees $n-1,n,n+1$ are isomorphisms by step 2.3, all of those degrees being at least $2$ because $n\ge 3$. The five lemma [F11] applied to the window ${\pi }_n(\tilde{L})\to {\pi }_n({\tilde{K}}_n)\to {\pi }_n({\tilde{K}}_n,\tilde{L})\to {\pi }_{n-1}(\tilde{L})\to {\pi }_{n-1}({\tilde{K}}_n)$ and to the window ${\pi }_{n+1}({\tilde{K}}_n)\to {\pi }_{n+1}(\tilde{K})\to {\pi }_{n+1}(\tilde{K},{\tilde{K}}_n)\to {\pi }_n({\tilde{K}}_n)\to {\pi }_n(\tilde{K})$ therefore gives isomorphisms of abelian groups $p_{\ast }:{\pi }_n({\tilde{K}}_n,\tilde{L})\to {\pi }_n(K_n,L)$ and $p_{\ast }:{\pi }_{n+1}(\tilde{K},{\tilde{K}}_n)\to {\pi }_{n+1}(K,K_n)$.

If the inclusion is a homotopy equivalence, the triple boundary is an isomorphism. In the triple sequence of [F12] with $X=K$, $B=K_n$ and $A=L$, exactness at ${\pi }_n(K_n,L)$ makes it surjective, because ${\pi }_n(K,L)$ is trivial by step 2.4 and the kernel of ${\pi }_n(K_n,L)\to {\pi }_n(K,L)$ is therefore all of ${\pi }_n(K_n,L)$. It is also injective: write $\delta$ for it and ${\mathrm{\partial }}^{\mathrm{\prime }}$ for the pair boundary ${\pi }_{n+1}(K,K_n)\to {\pi }_n(K_n)$, so that $\delta =j\circ {\mathrm{\partial }}^{\mathrm{\prime }}$ with $j:{\pi }_n(K_n)\to {\pi }_n(K_n,L)$; if $\delta z=0$ then ${\mathrm{\partial }}^{\mathrm{\prime }}z\in \ker j=\mathrm{im}({\pi }_n(L)\to {\pi }_n(K_n))$ by exactness of the pair $(K_n,L)$ at ${\pi }_n(K_n)$, say ${\mathrm{\partial }}^{\mathrm{\prime }}z=i_{\ast }y$, and composing with ${\pi }_n(K_n)\to {\pi }_n(K)$, which kills $\mathrm{im}{\mathrm{\partial }}^{\mathrm{\prime }}$ by exactness of the pair $(K,K_n)$ at ${\pi }_n(K_n)$, gives $i_{K,L\ast }(y)=0$, so that $y=0$ because ${\pi }_n(L)\to {\pi }_n(K)$ is an isomorphism and thus ${\mathrm{\partial }}^{\mathrm{\prime }}z=0$; then exactness of the pair $(K,K_n)$ at ${\pi }_{n+1}(K,K_n)$ writes $z=j_{\ast }^{\mathrm{\prime }}w$ for the map $j_{\ast }^{\mathrm{\prime }}:{\pi }_{n+1}(K)\to {\pi }_{n+1}(K,K_n)$, which by naturality of the pair sequences in the map of pairs $(K,L)\to (K,K_n)$ factors as the composite of ${\pi }_{n+1}(K)\to {\pi }_{n+1}(K,L)$ with ${\pi }_{n+1}(K,L)\to {\pi }_{n+1}(K,K_n)$ and is therefore zero because ${\pi }_{n+1}(K,L)=0$ by step 2.4; hence $z=0$ and $\delta$ is injective.

For either lifted pair, write $a={\tilde{k}}_0$ and let $A$ be its simply connected subspace ($\tilde{L}$ for $({\tilde{K}}_n,\tilde{L})$ and ${\tilde{K}}_n$ for $(\tilde{K},{\tilde{K}}_n)$). Define the right action by $c\cdot g:={\beta }_{{\gamma }_g}((T_{g^{-1}}{)}_{\ast }c),$ where ${\gamma }_g$ is any path in $A$ from $a$ to $T_{g^{-1}}a$ and ${\beta }_{{\gamma }_g}$ changes the basepoint back to $a$. Such paths exist, and [F21] makes the result independent of the path. This is a right action: applying $g$ and then $h$ applies $T_{h^{-1}}{T}_{g^{-1}}=T_{(gh{)}^{-1}}$ and concatenates the path from $a$ to $h^{-1}a$ with the image under $T_{h^{-1}}$ of the path from $a$ to $g^{-1}a$, a path from $a$ to $(gh{)}^{-1}a$; path-composition for $\beta$ gives $(c\cdot g)\cdot h=c\cdot (gh)$. The class construction in [F4] moves the marked boundary value through $A$; applying $T_{g^{-1}}$ transports that move, and any path used to define the translated cell class differs from the transported path by a loop in simply connected $A$, so [F21] identifies the based classes. Thus $c_{\tilde{e}}\cdot g$ is the basis class of the lift $T_{g^{-1}}\tilde{e}$. The corresponding change-of-basepoint shell lies in $A$, so relative Hurewicz sends this class to $u_{T_{g^{-1}}\tilde{e}}$. As the lifts form a free $\pi$-orbit [F2], each homotopy and homology basis set is a free $\pi$-orbit.

The relative Hurewicz homomorphisms of steps 2.2 and 3.1 are isomorphisms because they carry a free basis to a free basis, and for the triple $\tilde{L}\subset {\tilde{K}}_n\subset \tilde{K}$ the square with upper row $\delta :{\pi }_{n+1}(\tilde{K},{\tilde{K}}_n)\to {\pi }_n({\tilde{K}}_n,\tilde{L})$ and lower row ${\mathrm{\partial }}_{\ast }:H_{n+1}(\tilde{K},{\tilde{K}}_n)\to H_n({\tilde{K}}_n,\tilde{L})$, joined vertically by $h$, commutes: the triple boundary $\delta$ is the boundary of the pair $(\tilde{K},{\tilde{K}}_n)$ followed by the relative map of $({\tilde{K}}_n,\tilde{L})$ [F12], on disk models these are restriction to the boundary sphere followed by the induced map of pairs [F10], and $h$ is natural in based maps of pairs with $h([f])=f_{\ast }[D^m,S^{m-1}]$ [F13], so for a disk model $f:(D^{n+1},S^n)\to (\tilde{K},{\tilde{K}}_n)$ one has $h(\delta [f])=h([f{\mathrm{\mid }}_{S^n}])=(f{\mathrm{\mid }}_{S^n}{)}_{\ast }[S^n]={\mathrm{\partial }}_{\ast }(f_{\ast }[D^{n+1},S^n])={\mathrm{\partial }}_{\ast }h([f])$, the middle equality because $[D^{n+1},S^n]$ has homology boundary the positive sphere generator.

Choose one oriented lift ${\tilde{e}}_e$ for each relative cell $e$ and write $c_e:=c_{{\tilde{e}}_e}$. By step 4.1, $c_e\cdot g$ is the basis class of $T_{g^{-1}}{\tilde{e}}_e$; as $g$ varies these are exactly the lifts of $e$, once each. Therefore ${\pi }_n({\tilde{K}}_n,\tilde{L})={⨁}_e{c}_{e}R$ and ${\pi }_{n+1}(\tilde{K},{\tilde{K}}_n)={⨁}_{e^{\mathrm{\prime }}}{c}_{e^{\mathrm{\prime }}}R$ are finite free right $R$-modules on the chosen oriented characteristic cells, and the same holds for $H_n({\tilde{K}}_n,\tilde{L};\mathbb{Z})$ and $H_{n+1}(\tilde{K},{\tilde{K}}_n;\mathbb{Z})$ with the classes $u_e$.

With $F_r:=\tilde{L}\cup {\tilde{K}}^r$ one has $F_{n-1}=\tilde{L}$, $F_n={\tilde{K}}_n$ and $F_{n+1}=\tilde{K}$. Apply [F14] to integral homology of the universal-cover pair. Its differential $d_{n+1}$ is precisely the composite in the square of step 4.2: the triple boundary to $H_n(F_n,\tilde{L};\mathbb{Z})$ followed by the map to $H_n(F_n,F_{n-1};\mathbb{Z})=H_n({\tilde{K}}_n,\tilde{L};\mathbb{Z})$. Deck transformations commute with the integral connecting maps, so this differential is right $R$-linear on the free abelian groups indexed by lifted cells. Thus its matrix in one chosen lift per cell is exactly the matrix of ${\mathrm{\partial }}_{\ast }$; no second change of coefficients to $R$ is made.

Define the right action of $R$ on ${\pi }_n(K_n,L)$ and on ${\pi }_{n+1}(K,K_n)$ by $c\cdot g:=p_{\ast }(p_{\ast }^{-1}(c)\cdot g)$; this is a well-defined right action because $p_{\ast }$ is an isomorphism by step 3.2 and the cover-side action is the right action established in step 4.1. Thus $p_{\ast }$ is an isomorphism of right $R$-modules carrying the basis $\{c_e\}$ of step 5.1 to a basis of the base-side module. Hence ${\pi }_n(K_n,L)$ and ${\pi }_{n+1}(K,K_n)$ are finite free right $R$-modules on the chosen oriented characteristic cells: replacing the chosen lift ${\tilde{e}}_e$ by $T_h{\tilde{e}}_e$ replaces $c_e$ by $c_e\cdot h^{-1}$, and reversing the orientation replaces $c_e$ by $-c_e$, so the basis is determined only up to these factors $\pm g$, and such a change alters a representing matrix only by the corresponding change of basis.

The isomorphism $p_{\ast }$ of step 3.2 carries the chosen basis of the cover to the chosen basis of the base, and by step 6.1 the boundary of the statement is the image under $p_{\ast }$ of the triple boundary $\delta$ of step 4.2; hence the matrix of $\mathrm{\partial }:{\pi }_{n+1}(K,K_n)\to {\pi }_n(K_n,L)$ in the chosen bases is exactly the matrix of the relative cellular differential $d_{n+1}$ computed in step 5.2.

By steps 7.1 and 3.3 the triple boundary is represented in the chosen bases by $d_{n+1}$ and is an isomorphism whenever the inclusion is a homotopy equivalence. Since $R=\mathbb{Z}[\pi ]$, [F22] forces its finite free source and target ranks to be equal, so the representing matrix is square; the matrices of the isomorphism and its inverse are mutually inverse by the coordinate description of right-linear maps. In the degenerate case in which the pair has no relative cells both modules are the zero module and the empty matrix is invertible. ∎