Homological Atiyah–Hirzebruch spectral sequence

If $X$ is empty, $X_{+}$ is a point and every group in question is zero by [F1], proving all assertions in that case. Otherwise put $N=\dim X$. Define $D_{p,q}=h_{p+q}(X^p)$ and $E_{p,q}=h_{p+q}(X^p,X^{p-1})$. Let $i$ be skeletal inclusion, $j$ the pair map and $k$ the pair boundary. The pair exact sequences give $\mathrm{im}i=\ker j$, $\mathrm{im}j=\ker k$ and $\mathrm{im}k=\ker i$ at their respective positions. Their bidegrees are $(1,-1),(0,0),(-1,0)$, so this is an initial exact couple and [F2] constructs the claimed pages and differentials.

We verify the second page using the suspension-compatible cofiber boundary required by [F1]. Write $G=h_q(\ast )$. In column zero, $X_{+}^0$ is a finite wedge of $S^0$'s, so $E_{0,q}={⨁}_{e^0}G$. For $p\ge 1$, [F5] identifies the skeletal quotient with the finite wedge of oriented $p$-cell spheres; [F1] initially gives coordinates $E_{p,q}={⨁}_{e^p}G$ using the specified suspension. For the standard oriented disk pair $(D^p,S^{p-1})$, its boundary is an isomorphism onto $K=\ker (h_{p+q-1}(S^{p-1})\to h_{p+q-1}(\ast ))$. Indeed the inclusion of a chosen boundary point makes $h_m(S^{p-1})\to h_m(D^p)$ surjective in every degree, since the disk contracts to that point; the pair exact sequence proves the assertion. For $p\ge 2$, quotienting the chosen boundary point identifies $K$ with ${\tilde{h}}_{p+q-1}(S^{p-1})$: the split cofiber sequence $S^0\to (S^{p-1}{)}_{+}\to (S^{p-1},s_0)$ proves this, without identifying $(S^{p-1}{)}_{+}$ with a based wedge. Use an orientation-preserving identification of the boundary sphere and the specified suspension coordinates to view this disk boundary as an automorphism $a_{p,q}:G\to G$. For $p=1$, $K=\ker (G\oplus G\stackrel{+}{\to }G)$ and use the coordinate $g\mapsto (-g,g)$ in initial, terminal order to define the automorphism $a_{1,q}$.

For fixed $(p,q)$ put $n=p+q$. Then $N_{p,q}^r=k^{-1}(\mathrm{im}(h_{n-1}(X^{p-r})\to h_{n-1}(X^{p-1})))$: for $r>p$ the source skeleton is empty, so the image is zero and $N_{p,q}^{\mathrm{\infty }}=\ker k$. Likewise $B_{p,q}^r=j(\ker (h_n(X^p)\to h_n(X^{p+r-1})))$, and for $p+r-1\ge N$ this kernel is $\ker (h_n(X^p)\to h_n(X))$.

Naturality for each characteristic disk pair computes $jk$ cell by cell. For $p\ge 2$ its component at a $(p-1)$-cell is the attaching sphere followed by the quotient projection onto that cell. On the reduced kernel just used, this map acts by its degree, by [F6]. This also covers an attaching map that is not based: move its image of one chosen sphere point to the target sphere basepoint along a path and extend that homotopy by the point cofibration [F5]; homotopy invariance gives the same action on the kernel of collapse to a point. Its degree is the incidence number, unchanged by that homotopy. Thus the component is $[e^p:e^{p-1}]a_{p,q}$ in the initial coordinates. For $p=1$, naturality sends $(-a_{1,q}g,a_{1,q}g)$ to the initial and terminal vertices; when these coincide the sum is zero. The component is again the signed endpoint incidence times $a_{1,q}$. For $p=0$ the target is zero. Define $T_{0,q}={\mathrm{i}\mathrm{d}}_G$ and $T_{p,q}=T_{p-1,q}{a}_{p,q}$ recursively, and change every coordinate in column $p$ by $T_{p,q}$. The new differential component is $T_{p-1,q}([e^p:e^{p-1}]a_{p,q})T_{p,q}^{-1}=[e^p:e^{p-1}]{\mathrm{i}\mathrm{d}}_G$, since group homomorphisms commute with integer multiplication. Hence the row complex is isomorphic to cellular chains with coefficients $G$.

Taking homology of that row gives $E_{p,q}^2\cong H_p(X;h_q(\ast ))$ by [F3] and [F6]; negative columns are zero. Functoriality of the skeletal inclusions proves $F_p\subseteq F_{p+1}$ directly. For $p<0$ the source is zero by [F1], and for $p\ge N$ the map is the identity of $h_n(X)$. Thus these images really form a finite exhaustive filtration before [F4] is applied.

By exactness, $\ker k=\mathrm{im}(j:D_{p,q}=h_n(X^p)\to E_{p,q})$. Define $\mathrm{\Phi }:\ker k\to F_p{h}_n(X)\mathrm{/}{F}_{p-1}{h}_n(X)$ by choosing $x\in h_n(X^p)$ with $j(x)=e$ and sending $e$ to the image of $x$ in $h_n(X)$ modulo $F_{p-1}$. This is well defined because two such lifts differ by $\ker j=\mathrm{im}(h_n(X^{p-1})\to h_n(X^p))$, and it is surjective by the definition of $F_p$. Its kernel is $j(\ker (h_n(X^p)\to h_n(X)))$: one inclusion is immediate, and if the image of $x$ lies in $F_{p-1}$, choose $y\in h_n(X^{p-1})$ with the same image in $h_n(X)$; then $x-i(y)$ maps to zero in $h_n(X)$ and $j(x)=j(x-i(y))$. Thus step 1.3 gives $E_{p,q}^{\mathrm{\infty }}=\ker k\mathrm{/}j(\ker (h_n(X^p)\to h_n(X)))\cong F_p{h}_n(X)\mathrm{/}{F}_{p-1}{h}_n(X)$.

Steps 1.1, 3.1 and 4.1 give the asserted first and second pages, the bidegree of the differentials and the identification of the stable page with the associated graded of the finite filtration, which proves the theorem.