An identity relative homotopy matrix permits cell cancellation

Write $K_n=L\cup \{e_1^n,\dots ,e_a^n\}$. For each lower characteristic class $b_j\in {\pi }_n(K_n,L)$, the identity matrix supplies an upper class $u_j\in {\pi }_{n+1}(K,K_n)$ with triple boundary $\delta u_j=b_j$. The composite ${\pi }_{n+1}(K,K_n)\stackrel{\delta }{\to }{\pi }_n(K_n,L)\stackrel{\mathrm{\partial }}{\to }{\pi }_{n-1}(L)$ is zero by exactness of the triple/pair boundary construction. Thus $\mathrm{\partial }{b}_j=0$: the attaching sphere of each lower cell is null-homotopic in $L$.

Apply the finite homotopy-of-attaching-map collar of Cohen’s attaching-map comparison, printed p.23, to trivialize each lower attaching map, pushing upper maps along the induced deformation. This uses [F2] and [F3], and is the simplification established in Cell slides and stabilizations realize elementary group-ring matrices. The resulting lower skeleton has the form $K_n=L\vee {\bigvee }_{j=1}^a{S}_j^n$, and the relative homotopy matrix is still the identity after transporting its characteristic bases.

The relative inclusion of the wedge $L\vee \bigvee S_j^n$ has, in degree $n$, a split exact sequence ${\pi }_n(L)\to {\pi }_n(K_n)\to {\pi }_n(K_n,L)\to 0$: retraction $K_n\to L$ splits the first map, and the constant lower attaching maps make the last boundary zero. A relative basis vector $b_j$ therefore has a spherical representative ${\sigma }_j:S^n\to K_n$ that maps a chosen $n$-disk homeomorphically through the characteristic disk of $e_j^n$ and sends its complement to the basepoint in $L$. More generally, a class with zero $j$th relative coordinate has a representative avoiding the interior of $e_j^n$, since its $R$-linear sphere terms use only the other wedge summands and its residual term lies in ${\pi }_n(L)$.

Let ${\phi }_j:S^n\to K_n$ be the attaching map of the $j$th upper cell. Its relative image is $b_j$ by the identity-matrix hypothesis, while ${\sigma }_j$ has the same image. Exactness in step 3.1 gives $[{\phi }_j]-[{\sigma }_j]\in \mathrm{im}{\pi }_n(L)$; represent this difference by a based sphere ${\alpha }_j$ in $L$ and pinch it into the complementary disk of ${\sigma }_j$. The resulting map ${\sigma }_j+{\alpha }_j$ is homotopic to ${\phi }_j$ through maps to $K_n$ and still maps one prescribed disk homeomorphically onto the lower $j$th cell with every other point outside that cell. This is the homotopy-level correction in Cohen’s identity-matrix cancellation, printed p.30.

For $i\mathrm{\ne }j$, the identity matrix gives zero $j$th relative coordinate to ${\phi }_i$. By the last clause of step 3.1, homotope ${\phi }_i$ to an attaching map missing the interior of $e_j^n$. Replace the upper attaching maps by these homotopic representatives, one at a time, via finite collar expansions and collapses relative to $K_n$; [F2] transports subsequent attachments. Afterward $e_j^n$ occurs in the boundary of exactly the corrected upper cell $e_j^{n+1}$, and the corrected ${\phi }_j$ meets it on one disk by a homeomorphism. It is now a genuine free face and [F3] removes the pair.

The remaining pair still has only $n$- and $(n+1)$-cells, and its triple boundary in the remaining transported bases is the identity with row and column $j$ deleted: the other corrected upper maps have no $e_j^n$ term and the first upper map is gone. Induct on the finite common number $a$; for $a=0$ the pair already equals $L$, while step 5.1 reduces $a$ by one. The finite concatenation of relative elementary moves proves the claim. ∎