Cellular reduction for a highly connected pair

By [F1] obtain $Q:(Z,A)\to (X,A)$, equal to the identity on $A$, with no relative cells below $n$ and an inverse rel $A$ under [A1]. The maps and homotopies induce inverse maps on relative homotopy and relative homology: based relative representatives compose with the homotopies fixing $a$; on relative chains, [F6]'s prism identity descends because every prism over a simplex in $A$ remains in $A$, and therefore makes the two composites induce the identity in relative homology. These maps commute with Hurewicz by its naturality. It suffices to calculate on this model. In the rest of the proof $Z_{n-1}=A$ and $Z_k$ contains all of $A$, including its cells above dimension $k$.

For any triple $A\subseteq B\subseteq T$, its integral relative chain groups give a short exact sequence $0⟶C_{\bullet }(B)\mathrm{/}{C}_{\bullet }(A)⟶C_{\bullet }(T)\mathrm{/}{C}_{\bullet }(A)⟶C_{\bullet }(T)\mathrm{/}{C}_{\bullet }(B)⟶0.$ Indeed the first arrow is injective since a chain in $B$ that lies in $C_{\bullet }(A)$ is already zero in its source quotient; the second is surjective by taking the same representative; its kernel consists exactly of the images of chains in $B$. The boundary maps commute with these quotients. By [F5] we therefore have the homology triple sequence. Its connecting map sends the class of a chain $c$ with $\mathrm{\partial }c\in C_{\bullet }(B)$ to the relative class of $\mathrm{\partial }c$ modulo $C_{\bullet }(A)$. This formula is well defined: changing $c$ by a $B$ chain adds a relative boundary in $(B,A)$, and changing it by a boundary adds zero; it is the usual lift-then-boundary connecting construction in [F5].

For $k\ge n$, $Z_k\mathrm{/}{Z}_{k-1}$ is the wedge of its $k$-cell quotient spheres by [F4]. The subspace $Z_{k-1}$ is nonempty. Thus quotient comparison, the positive point-relative identification [F6], and wedge homology [F4] give $H_i(Z_k,Z_{k-1})=0(i\mathrm{\ne }k),H_k(Z_k,Z_{k-1})={⨁}_{E_k}\mathbb{Z}\mathrm{.}$ In the displayed vanishing $i\ge 0$: for $i=0$ the quotient wedge is path connected and the pair sequence in [F5] identifies its point-relative $H_0$ with the zero cokernel of $\mathbb{Z}\to \mathbb{Z}$. The generator in each summand is the characteristic disk's relative orientation image, as follows from the quotient orientation convention and its commuting characteristic maps in [F3]. This homology calculation does not require simple connectivity of $A$ or $Z_{k-1}$.

Put $Y=Z_{n+1}$. The pair $(Z,Y)$ has relative cells only in dimensions at least $n+2$, so [F2] gives ${\pi }_i(Z,Y,a)=0$ for $1\le i\le n+1$. The triple segment for $A\subseteq Y\subseteq Z$ is therefore $0⟶{\pi }_n(Y,A,a)⟶{\pi }_n(Z,A,a)⟶0.$ Exactness makes the middle map surjective with zero kernel, hence an isomorphism, also when $n=2$ and relative groups are not assumed abelian. This proves the homotopy stability for arbitrary $A$.

Now assume $A$ is simply connected and put $B=Z_n$. By [F2], $B$ is path connected and ${\pi }_1(A,a)\to {\pi }_1(B,a)$ is surjective, since its relative cells have dimension $n\ge 2$. Hence $B$ is simply connected. Apply [F3] first to $(B,A)$ with its $n$-cells and then to $(Y,B)$ with its $(n+1)$-cells. Write $c_{\alpha },u_{\alpha }$ for the first homotopy and homology bases, and $c_{\beta },u_{\beta }$ for the second. Both pairs of groups are free abelian, and Hurewicz sends each $c$ to the corresponding $u$. The bases use orientations of the actual disks and their boundaries, not a sign inferred from a numerical rank.

Apply step 1.2 to $(Z_k,Z_{k-1},A)$. Step 2.1 and induction starting with $H_i(A,A)=0$ show $H_i(Z_k,A)=0$ for $0\le i<n$. If $k\ge n+2$, the two terms $H_{n+1}(Z_k,Z_{k-1})$ and $H_n(Z_k,Z_{k-1})$ both vanish, so exactness gives an isomorphism $H_n(Z_{k-1},A)\to H_n(Z_k,A)$. Every relative cycle in $(Z,A)$ is represented by a finite ordinary chain by [F5]; [F8] puts its finitely many compact simplex images in a finite CW subcomplex, hence in some $Z_k$ after adjoining $A$. The same holds for a bounding chain in an equation $c=\mathrm{\partial }d+a^{\mathrm{\prime }}$ with $a^{\mathrm{\prime }}\in C_{\bullet }(A)$. Thus every class comes from a finite stage, and any equality becomes valid at a finite stage: chain inclusion is injective on the free simplex generators. Finite-stage lower vanishing proves $H_i(Z,A)=0$ for $i<n$. For degree $n$, surjectivity from $Z_{n+1}$ follows by moving a finite-stage class backward through the isomorphisms just proved. If a class from $Z_{n+1}$ dies in $Z$, its bounding chain lies in a finite stage where these same isomorphisms prove it was already zero. This proves the asserted homology stability without an infinite-stage exactness assumption.

The pair $(Y,B)$ is $n$-connected by [F2]. Its relative degree-$n$ group is zero. The homotopy triple segment for $A\subseteq B\subseteq Y$ gives ${\pi }_{n+1}(Y,B,a)\stackrel{{\delta }_{\pi }}{\to }{\pi }_n(B,A,a)⟶{\pi }_n(Y,A,a)⟶0.$ The middle group is free abelian by step 2.3, so its surjective image is abelian as well, including for $n=2$. Exactness identifies the target with the abelian cokernel of ${\delta }_{\pi }$. Likewise step 2.1 gives $H_n(Y,B)=0$, so the homology triple sequence in step 1.2 gives $H_{n+1}(Y,B)\stackrel{{\delta }_H}{\to }{H}_n(B,A)⟶H_n(Y,A)⟶0.$ It identifies this target with the cokernel of ${\delta }_H$. All maps shown are actual inclusion or boundary maps, and the two left groups have the respective cell bases from step 2.3.

Fix one $(n+1)$-cell $\beta$. Its characteristic map can be homotoped as a map of pairs $(D^{n+1},S^n)\to (Y,B)$ to a based representative ${\chi }_{\beta }^{\mathrm{\prime }}$ of $c_{\beta }$, by the explicit marked-point HEP construction in [F3]. Its boundary sphere map ${\phi }_{\beta }^{\mathrm{\prime }}$ is based at $a$ and homotopic in $B$ to the original attaching map ${\phi }_{\beta }$, possibly through a moving-basepoint homotopy. By the definition of the homotopy triple boundary in [F2], ${\delta }_{\pi }(c_{\beta })$ is the image of $[{\phi }_{\beta }^{\mathrm{\prime }}]$ in ${\pi }_n(B,A,a)$. Under the quotient and wedge homotopy basis identification [F3, F4], its $\alpha$ coefficient is $\mathrm{deg}(p_{\alpha }{q}_n{\phi }_{\beta }^{\mathrm{\prime }})$. Homotopy invariance in [F7] makes this equal to $D_{\alpha \beta }=\mathrm{deg}(p_{\alpha }{q}_n{\phi }_{\beta })$. These coefficients have finite support: the compact attaching sphere meets finitely many cells, so its quotient image meets only finitely many $n$-sphere summands, by [F8]. Outside those summands the projected map is constant and its degree is zero. Hence these columns define a homomorphism between the displayed direct sums.

Represent $[D^{n+1},S^n]$ by a relative orientation chain $d$ with $\mathrm{\partial }d$ an absolute sphere cycle representing the positive boundary orientation; this is precisely the convention of [F6]. The characteristic image ${\chi }_{\beta \mathrm{\#}}d$ represents $u_{\beta }$. The connecting formula of step 1.2 sends it to $[{\phi }_{\beta \mathrm{\#}}\mathrm{\partial }d]$ in $H_n(B,A)$. This sign is positive because boundary is taken before projection to the relative quotient; no reordering or sign convention for a double complex enters. Apply $q_n$ and then $p_{\alpha }$. The resulting class is $(p_{\alpha }{q}_n{\phi }_{\beta }{)}_{\ast }[S^n]$, whose coefficient on the target orientation is $D_{\alpha \beta }$ by [F7]. The homology basis and its coordinate inverse in [F3, F4] therefore give ${\delta }_H(u_{\beta })={\sum }_{\alpha }{D}_{\alpha \beta }{u}_{\alpha }\mathrm{.}$ The sums are finite by step 4.1. Comparing with that step and $h(c)=u$ from step 2.3 shows $h{\delta }_{\pi }={\delta }_{H}h$ on every generator and therefore on every finite sum.

Steps 4.1–5.1 identify both arrows in step 3.2 with the same integer matrix $D$. Naturality of Hurewicz [F6] makes the square of inclusion maps from $(B,A)$ to $(Y,A)$ commute. Since those maps are surjective, the induced map on their cokernels is precisely $h:{\pi }_n(Y,A,a)\to H_n(Y,A)$: every target class is represented by a finite sum of the $c_{\alpha }$, on which $h$ has the stated formula. Thus the identifications are with the actual Hurewicz map. By steps 2.2 and 3.1 the same description holds after all higher cells are attached, and by step 1.1 it transports through the equivalence rel $A$ to the original pair.

Empty cell sets give zero free groups, zero columns and the usual zero-map cokernel, all retained in the formulas. If $A=X$, the chosen model may still add cells, but step 1.1 identifies its relative groups with zero and the same cokernel proof applies. If $A$ is a point or one cell is attached, the quotient and single-basis calculations above still apply. The lower homology range includes zero by step 2.1; no relative homotopy degree zero or one is asserted here. At $n=2$, step 2.3 supplies abelianness before any abelian cokernel is taken. Without simple connectivity only the model, lower homology vanishing and stability conclusions are asserted; the free-basis assertion was used only after that extra hypothesis. All operations on a supplied model use individual finite chains, individual homotopies or unique coordinate formulas, without infinite choice. The sole AC-dependent supplier in step 1.1 uses AC for arbitrary-cell cellular approximation and simultaneous compression-disk selection as stated in [F1], and that assumption propagates to the claimed equivalence. These checks complete every assertion.