Cell trading puts a finite relative equivalence in two high degrees

Since the inclusion is a homotopy equivalence, it induces isomorphisms ${\pi }_r(L)\to {\pi }_r(K)$ for every $r\ge 1$ and a bijection ${\pi }_0(L)\to {\pi }_0(K)$; exactness of [F3] at ${\pi }_r(K,L)$ for $r\ge 1$ then forces ${\pi }_r(K,L)=0$, and connectivity of $K$ together with $L\mathrm{\ne }\mathrm{\varnothing }$ makes every vertex of $K$ the endpoint of a path in $K$ from a vertex of $L$.

For $r=0$ let $e^0$ be a relative vertex; the path of step 1.1 starting at $e^0$ is a homotopy $H:D^0\times I\to K$ with $H_0$ the identity on the point and $H_1(D^0)\subseteq L$, the boundary condition being vacuous.

Suppose all relative cells of $K$ outside $L$ have dimension at least $r$, and let $e^r$ be a relative $r$-cell with attaching map $\phi :S^{r-1}\to K^{(r-1)}=L^{(r-1)}$. For $r\ge 2$, injectivity of ${\pi }_{r-1}(L)\to {\pi }_{r-1}(K)$ makes $\phi$ null-homotopic in $L$, so choose a filling $\mathrm{\Psi }:D^r\to L$ with boundary $\phi$. The sphere obtained by gluing $\mathrm{\Phi }$ to the reverse of $\mathrm{\Psi }$ represents a class of ${\pi }_r(K)$; surjectivity of ${\pi }_r(L)\to {\pi }_r(K)$ permits changing $\mathrm{\Psi }$ by a sphere map in $L$ until this glued sphere is null. The resulting null-homotopy is precisely a homotopy $H:D^r\times I\to K$ from $\mathrm{\Phi }$ into $L$ fixing its entire boundary. For $r=1$, choose a path in connected $L$ between the endpoints of $\mathrm{\Phi }$ and use the isomorphism ${\pi }_1(L)\to {\pi }_1(K)$ to homotope the two paths rel endpoints; the separate $r=0$ case is handled by the vertex path.

Put $W=L\cup \overline{e^r}$, with its actual CW structure inherited from $K$. Extend the homotopy of step 2.2 (or the vertex path of step 2.1) by the identity on $L$ to a map $\widehat{H}:W\times I\to K$; it descends through the attaching identifications because the boundary track is fixed. Its endpoint is a retraction $a:W\to L$. Apply [F4] to $a$ relative to $L$, obtaining a cellular $a^{\mathrm{\prime }}:W\to L$ and a homotopy rel $L$ from $a$ to $a^{\mathrm{\prime }}$. Concatenate with $\widehat{H}$, and apply the cellular-homotopy clause of [F4] to the maps $W\hookrightarrow K$ and $W\stackrel{a^{\mathrm{\prime }}}{\to }L\hookrightarrow K$, relative to $L$. Both endpoint maps and the fixed $L$ track are cellular. Restriction along the characteristic map $\mathrm{\Phi }:D^r\to W$ now gives a homotopy $H$ with $H_0=\mathrm{\Phi }$, $H_t{\mathrm{\mid }}_{\mathrm{\partial }{D}^r}=\phi$, $H_1(D^r)\subseteq L^{(r)}$ and $H(D^r\times I)\subseteq K^{(r+1)}$, since $\mathrm{\Phi }(D^r)\subseteq W^{(r)}$. Its boundary lies in $L\cup \overline{e^r}$. This applies approximation on $W$, not on a sphere whose attaching map might be noncellular.

Write $Q=D^r\times I$, an $(r+1)$-ball. Attach an $(r+1)$-cell $a$ to $K$ by $H{\mathrm{\mid }}_{\mathrm{\partial }Q}$, whose image lies in $(L\cup \overline{e^r})\cap K^{(r)}$ by the endpoint and boundary bounds of step 3.1. Then attach an $(r+2)$-cell using a boundary sphere written as two $(r+1)$-disks glued along their boundary: map one disk by $H:Q\to K^{(r+1)}$ and the other by the characteristic map of $a$, with matching boundary parameterizations. This defines a CW complex $M$ and an elementary expansion $K\hookrightarrow M$, with $a$ as its free face. The subspace $L\cup \overline{e^r}\cup \bar{a}$ is a subcomplex. Denote $a$ by $e^{r+1}$ below.

In the subcomplex $C:=L\cup \overline{e^r}\cup \overline{e^{r+1}}\subseteq M$ the cell $e^r$ is a free face of $e^{r+1}$: the attaching map of $e^{r+1}$ restricts on the face $D^r\times \{0\}$ to the characteristic map $\mathrm{\Phi }$ of $e^r$, a homeomorphism from the open disk onto $e^r$, and maps the complementary part $\mathrm{\partial }{D}^r\times I\cup D^r\times \{1\}$ into $L$ by step 3.1; no other cell of $C$ has the interior of $e^r$ in its closure, since $e^r\mathrm{\notin }L$ and $e^{r+1}$ is the only other cell of $C$ outside $L$. Hence $C\searrow L$ is an elementary collapse and $C$ is an elementary expansion of $L$ of dimension $r+1$.

By [F5] and [F10] the pair $(C,L)$ admits a strong deformation retraction $G:C\times I\to C$ with $G_0={\mathrm{i}\mathrm{d}}_C$, $G_1(C)\subseteq L$ and $G_t{\mathrm{\mid }}_L={\mathrm{i}\mathrm{d}}_L$. First apply [F4] to the endpoint retraction $G_1:(C,L)\to (L,L)$ to obtain a cellular map $g:C\to L$ homotopic to $G_1$ relative to $L$. The subspace $A=C\times \{0\}\cup L\times I\cup C\times \{1\}$ is a CW subcomplex of $C\times I$, so its inclusion is a cofibration by [F10]. Extend the endpoint homotopy from $G_1$ to $g$ across $C\times I$ by HEP while retaining the bottom identity and the fixed $L\times I$ track. This produces a deformation from ${\mathrm{i}\mathrm{d}}_C$ to $g$, fixed on $L$, whose restriction to all of $A$ is cellular. Now apply [F4] to this prism map relative to $A$ to make the entire homotopy cellular without changing its bottom, side or top. With the product CW structure, $C^{(m)}\times I$ lies in the $(m+1)$-skeleton of $C\times I$, so the resulting homotopy satisfies $G(C^{(m)}\times I)\subseteq C^{(m+1)}$; its endpoint $G_1=g$ satisfies $G_1(C^{(m)})\subseteq L^{(m)}$ for every $m$. The later push uses both this endpoint bound and the $+1$ prism bound: if an attaching sphere lands in $C^{(k-1)}$, its side track lands in $C^{(k)}$. No degree-$m$ bound on the full track is asserted.

Push claim. Let ${\phi }_0:S^{k-1}\to C^{(k-1)}$ be any map, put $X:=C{\cup }_{{\phi }_0}{D}^k$ with characteristic map ${\mathrm{\Phi }}_0$ of its new cell, and put $Z:=L{\cup }_{G_1{\phi }_0}{D}^k$ with characteristic map ${\mathrm{\Phi }}_1$; then $X$ and $Z$ are related by finitely many elementary expansions and collapses. Indeed $Y$ is the complex $C$ with the cell attached along $G_1{\phi }_0$ and $Z$ is the complex $L$ with that cell attached, so $Y$ is obtained from $Z$ by adding back the elementary expansion pair; moreover ${\mathrm{\Phi }}_1{\mathrm{\mid }}_{S^{k-1}}=G_1{\phi }_0$ matches $G({\phi }_0(x),t)$ at $t=1$.

In the situation of step 7.1 build $J$ from $X$ by attaching a further $k$-cell $\widehat{e}$ along $G_1{\phi }_0$ and a $(k+1)$-cell $\mathrm{\Pi }$ whose boundary disk $\mathrm{\partial }(D^k\times I)$ is glued by the usual three pieces: the face $D^k\times \{0\}$ by ${\mathrm{\Phi }}_0$, the face $D^k\times \{1\}$ by the characteristic map of $\widehat{e}$, and the side $S^{k-1}\times I$ by $(x,t)\mapsto G({\phi }_0(x),t)$, which is legitimate as a CW attaching map because ${\phi }_0(S^{k-1})\subseteq C^{(k-1)}$ and the $+1$ bound of step 6.1 puts its side track in $C^{(k)}\subseteq X^{(k)}$; the two end values agree with the corresponding face maps. Then $X\hookrightarrow J$ is an elementary expansion of dimension $k+1$ with free face $\widehat{e}$, and likewise $Y\hookrightarrow J$ is an elementary expansion of dimension $k+1$ with free face the $k$-cell of $X$; both use [F1], the side values lying in $C$.

In $Y$ the closure of the new $k$-cell meets $C$ exactly in $G_1{\phi }_0(S^{k-1})\subseteq L$, so it is disjoint from the interior of $e^r$; therefore the interior of $e^r$ lies in the closure of no cell of $Y$ other than $\overline{e^r}$ and $\overline{e^{r+1}}$, and the elementary collapse of step 5.1 is still available in $Y$: $Y\searrow Z$. Reading the move of step 8.1 followed by the collapse just constructed gives a formal deformation $X\hookrightarrow J\searrow Y\searrow Z$, so $X$ and $Z$ are related by elementary expansions and collapses.

Transport along a deformation. If $D_0,\dots ,D_m$ is a formal deformation of finite CW complexes in which each collapse step admits the cellular retraction data of step 6.1, and $\phi :S^{k-1}\to D_0^{(k-1)}$ is an attaching map, then $D_0{\cup }_{\phi }{D}^k$ is related by elementary expansions and collapses to $D_m{\cup }_{\psi }{D}^k$, where $\psi$ is obtained from $\phi$ by composing with the cellular inclusions of the expansion steps and the time-one maps of the collapse steps. This is proved by induction on $m$: a collapse step is step 9.1 applied with $C:=D_0$ and $L:=D_1$, an expansion step changes no attaching data because $D_0{\cup }_{\phi }{D}^k\subseteq D_1{\cup }_{\phi }{D}^k$ differs only by the expansion pair, and the induction hypothesis is then applied to the remaining steps with the transported attaching map, which is legitimate because the time-one maps are cellular on the complex they contract.

Trading one cell. Take $C=L\cup \overline{e^r}\cup \overline{e^{r+1}}$ as in step 5.1 and attach the relative cells of $K$ other than $e^r$ in their original CW order, followed by the new $(r+2)$-cell of $M$. This is a legitimate relative CW filtration over $C$: every old cell's attaching image lies in the earlier old skeleta (now including $C$), and the last cell attaches by $H(D^r\times I)\subseteq K^{(r+1)}$, which is present by then. It is not asserted that the other old relative cells of degrees $r$ and $r+1$ lie in $C$, nor that the new last cell attaches to $C$ alone. Repeatedly applying step 10.1 to push each attaching map across the collapse $C\searrow L$ produces a formal deformation from $M$ to a complex $K^{\mathrm{\prime }}:=L\cup d_1^{\mathrm{\prime }}\cup \cdots \cup d_s^{\mathrm{\prime }}$ in which each $d_j^{\mathrm{\prime }}$ has the same dimension as the corresponding old cell, and the final new cell has dimension $r+2$. This is the finite push construction in Cohen’s cell-trading construction, printed pp.25–26.

Consequences for the trading step. By steps 4.1 and 11.1 the complexes $K$ and $K^{\mathrm{\prime }}$ are related by finitely many elementary expansions and collapses, and every move is performed relative to the cells retained by the previous steps and fixes $L$; the relative cells of $K^{\mathrm{\prime }}$ over $L$ are the relative cells of $K$ other than $e^r$, each with the same dimension, together with one cell of dimension $r+2$. Hence $K^{\mathrm{\prime }}$ has one relative $r$-cell fewer than $K$, has no relative cells of dimension below $r$, and agrees with $K$ in the number of relative cells in every degree other than $r$ and $r+2$.

For the iteration, let $K^{(1)}$ be a finite CW complex containing $L$ and related to $K$ by a formal deformation fixing $L$; the composite of the moves restricts to the identity on $L$ and is a homotopy equivalence $f:K\to K^{(1)}$, so applying step 1.1 to the pairs $(K,L)$ and $(K^{(1)},L)$ and using exactness of [F3] at ${\pi }_r(K^{(1)},L)$ for $r\ge 1$ gives ${\pi }_r(K^{(1)},L)=0$ as well.

Iterating step 12.1 for $r=0,1,2$ using the data of steps 2.1, 2.2 and 13.1 removes all relative cells of dimension at most two and yields a finite CW complex $K^{(1)}\supseteq L$, related to $K$ by finitely many elementary expansions and collapses relative to $L$, all of whose relative cells have dimension at least three.

Choose once and for all an integer $n\ge \max (4,\dim K^{(1)}+1)$. For each $r=3,4,\dots ,n-1$ in this finite list, apply step 12.1 to every relative $r$-cell then present. Each trade deletes one $r$-cell and creates only an $(r+2)$-cell, so no later trade creates a cell in a degree already processed. By step 13.1 the required relative homotopy groups remain zero. At the end no relative cell has degree below $n$, while every old cell had degree at most $\dim K^{(1)}<n$ and every created cell has degree at most $n+1$. Thus the only possible relative degrees are $n,n+1$, exactly the fixed-target argument of Cohen’s two-layer reduction, printed pp.26–27.

Torsion transport. The composite $f:K\to K^{\mathrm{\prime }}$ of the moves of the deformation is a finite composite of elementary expansions, elementary collapses and identities between finite CW complexes, hence a simple homotopy equivalence by [F6] and satisfies $\tau (f)=0$ in $\mathrm{W}\mathrm{h}({\pi }_1{K}^{\mathrm{\prime }})$ by [F7]; writing $i:L\hookrightarrow K$ and $i^{\mathrm{\prime }}:L\hookrightarrow K^{\mathrm{\prime }}$ for the inclusions, $i^{\mathrm{\prime }}=f\circ i$ and [F8] give $\tau (i^{\mathrm{\prime }})=\tau (f)+f_{\ast }\tau (i)=f_{\ast }\tau (i)$, so the relative torsion is transported by $f_{\ast }$.

By steps 14.1, 15.1 and 16.1 the finitely many elementary expansions and collapses constructed above carry $(K,L)$ to a pair $(K^{\mathrm{\prime }},L)$ whose relative cells lie only in two adjacent degrees $n,n+1$ with $n\ge 3$, the deformation fixes $L$ and hence respects the homotopy class of the inclusion, and the relative torsion is transported along it; the cells of dimension $0$ and $1$ were removed in the first iteration using the connectedness data of step 2.1 and the ${\pi }_1$-isomorphism of step 1.1. ∎