Finite additivity alone does not prove infinite cw uniqueness

For a family of abelian groups $M_i$, denote ${\prod }_{i\ge 0}{M}_i\mathrm{/}{⨁}_{i\ge 0}{M}_i$ by $Q(M)$. Degreewise maps induce maps of $Q$, giving functoriality. If $A_i\stackrel{u_i}{\to }{B}_i\stackrel{v_i}{\to }{C}_i$ is exact at every $B_i$, then $Q(A)\to Q(B)\to Q(C)$ is exact: when $v(b)$ is finitely supported, replace the finitely many bad coordinates of $b$ by zero; the resulting equivalent tuple lies in $\prod \ker v_i$. Choose coordinate lifts in $A_i$ and obtain a preimage in $Q(A)$. The converse inclusion follows from $v_i{u}_i=0$. This uses ordinary choice for the countable family of nonempty lift sets. Adding, deleting, or altering finitely many initial coordinates has no effect on $Q$.

Apply step 1.1 to the ordinary singular pair sequence supplied by [F2]. Define the $E$ boundary by sending a representative $(x_i{)}_{i\ge 0}$ to the tuple whose $j$th coordinate is the singular boundary $\mathrm{\partial }{x}_{j+1}\in H_j(A;\mathbb{Z})$. Dropping the unused initial coordinate is precisely the shift identification of step 1.1. At each term the coordinatewise singular exact sequence and that shift prove the required exactness, even though $E_n$ is independent of $n\in \mathbb{Z}$. Homotopies and excision induce coordinatewise equal maps and isomorphisms, so they do so on $Q$. Apply [F1] here only to the ordinary singular theory [F2]: its quotient identifications and suspension isomorphisms give coordinatewise reduced cofiber sequences and suspension maps. Applying $Q$ and step 1.1 gives the reduced $E$ axioms and identifies them with the displayed pair construction. Replacing reduced absolute groups by unreduced absolute groups changes only coordinate zero and therefore does not change $Q$. All structure maps are natural.

A point, the empty space under the zero reduced-empty convention, and $S^0$ have reduced homology in at most one degree. Their $Q$ groups vanish; hence the coefficient of $E$ is zero. For any finite family, the product over degrees commutes with its finite direct sum, as does the finite-support subgroup. Thus finite disjoint-sum additivity holds for pairs and finite wedge additivity for reduced spaces. For a finite-dimensional CW pair of dimension at most $N$, [F3] and [F4] applied to singular homology give $H_i(X,A;\mathbb{Z})=0$ outside $0\le i\le N$, regardless of the number of cells. Therefore $E_n(X,A)=0$ for every integer $n$.

Give $W={\bigvee }_{i\ge 1}{S}^i$ its CW structure with one vertex and one cell in each positive dimension, all attached by constant maps. The cellular differentials vanish, including the degree-one terminal-minus-initial differential. On each skeleton [F3] and [F4] compute one copy of $\mathbb{Z}$ in each positive degree present. Applying [F5] to ordinary singular homology, not to $E$, gives ${\tilde{H}}_i(W;\mathbb{Z})=\mathbb{Z}$ for every $i\ge 1$ and zero in degree zero. Hence $E_n(W)={\prod }_{i\ge 1}\mathbb{Z}\mathrm{/}{⨁}_{i\ge 1}\mathbb{Z}\mathrm{\ne }0$: the all-ones tuple is not finitely supported. Yet each $E_n(S^i)=0$ by step 2.2, so the canonical infinite wedge map from the direct sum of these zero groups is not surjective.

Taking the direct sum of the pair functors $H$ and $E$, with componentwise structure maps, preserves homotopy, exactness, excision, dimension, and finite additivity. Its coefficient at a point is $\mathbb{Z}\oplus 0=\mathbb{Z}$, and step 2.2 identifies it with $H$ on all finite-dimensional pairs. On $W$, ordinary $H_{-1}(W;\mathbb{Z})=0$ by the nonnegative singular chain complex, whereas $(H\oplus E{)}_{-1}(W)=E_{-1}(W)\mathrm{\ne }0$ by step 3.1. Thus even an abstract graded-group equivalence fails there. The ordinary all-CW uniqueness theorem requires arbitrary additivity, exactly the axiom that this construction fails.