The cohomology universal coefficient sequence splits nonnaturally

Choose the cycle projection ${\pi }_n:C_n\to Z_n$ from [F2] and let $q:Z_n\to H_n$ be the quotient. For $u\in W$, define $s(u)=[uq{\pi }_n]$. It is a cocycle because $q{\pi }_n$ kills $B_n$; restriction to $Z_n$ is $uq$, so $\beta s(u)=u$. Additivity follows from $(u+v)q{\pi }_n=uq{\pi }_n+vq{\pi }_n$.

Define $T:E\oplus W\to V$ by $T(e,u)=\iota e+s(u)$. If it is zero, apply $\beta$ to obtain $u=0$, then injectivity of $\iota$ gives $e=0$. For any $v\in V$, the element $v-s(\beta v)$ has zero evaluation, so exactness gives a unique $e\in E$ with $\iota e=v-s(\beta v)$. Thus $v=T(e,\beta v)$. The inverse is additive because this equation determines $e$ uniquely and all its other maps are additive. Hence $T$ is the claimed isomorphism.

If ${\pi }_n^{\mathrm{\prime }}$ is another cycle projection, both sections have the same composite with $\beta$, so their difference has image in $\iota E$. For each $u$, the difference is the class $[uq({\pi }_n-{\pi }_n^{\mathrm{\prime }})]$. No argument above makes this class zero, nor supplies projections compatible with all maps of spaces. Thus the constructed direct-sum map depends on the supplied projections; a failure of natural splitting is not inferred merely from this dependence. The separate counterexample page addresses nonexistence. For fixed $C$ and fixed ${\pi }_n$, coefficient postcomposition does commute with $s$ by its formula.

At $n=0$, $E=0$ and $\beta$ is an isomorphism, so its inverse is the unique section. For an empty space, equal pair $A=X$, or $G=0$, all groups are zero; for a point in degree zero, $T$ identifies $G$ with $\mathrm{Hom}(\mathbb{Z},G)$ by evaluation at $1$. AC enters through [F2] to supply the projections for arbitrary-rank chains and is already assumed in [F1]. The inverse construction in step 2.1 uses unique elements, not a further choice function.