The homology Kunneth sequence splits nonnaturally

The singular complexes are nonnegative free PID complexes, so the section theorem [F2] applies. Define $j^{\mathrm{\prime }}=sj:Q\to W$ and $r^{\mathrm{\prime }}=rt:W\to K$. Then $\gamma j^{\mathrm{\prime }}=\beta tsj=\beta j=1$, $r^{\mathrm{\prime }}{\alpha }^{\mathrm{\prime }}=rts\alpha =1$, and $r^{\mathrm{\prime }}{j}^{\mathrm{\prime }}=rtsj=0$. All maps are $R$-linear.

The remaining composite satisfies $j^{\mathrm{\prime }}\gamma =sj\beta t=s(1-\alpha r)t=1-{\alpha }^{\mathrm{\prime }}{r}^{\mathrm{\prime }}$. Therefore $(k,q)\mapsto {\alpha }^{\mathrm{\prime }}k+j^{\mathrm{\prime }}q$ and $w\mapsto (r^{\mathrm{\prime }}w,\gamma w)$ are inverse: one composite is $(k,q)\mapsto (k,q)$ using $r^{\mathrm{\prime }}{j}^{\mathrm{\prime }}=0$, $\gamma {\alpha }^{\mathrm{\prime }}=0$ and the identity composites; the other is $w\mapsto ({\alpha }^{\mathrm{\prime }}{r}^{\mathrm{\prime }}+j^{\mathrm{\prime }}\gamma )w=w$. This proves the direct-sum assertion with the actual cross product and quotient.

The maps $r,j$ of [F2] use chosen retractions $C_p\to Z_{p}C$ and $D_q\to Z_{q}D$, obtained by splitting the surjections onto $B_{p-1}C$ and $B_{q-1}D$. No splitting of the inclusions $B_{p}C\hookrightarrow Z_{p}C$ is used. Since no compatibility of those retractions with all space maps is supplied, steps 1.1 and 2.1 assert existence of a section, without asserting its naturality. The canonical exact sequence itself retains its naturality from [F1]. This observation alone is not a proof that every possible natural section is impossible.

At $n=0$, $Q=0$ and $j^{\mathrm{\prime }}$ is the unique map from zero; ${\alpha }^{\mathrm{\prime }}{r}^{\mathrm{\prime }}=1$ gives the degree-zero cross-product isomorphism. If a factor is empty, $K=V=W=Q=0$ and every formula remains valid. If $K=0$, the section is the inverse of $\gamma$; if $Q=0$, no Tor summand is added. In degree zero on a pair of points, ${\alpha }^{\mathrm{\prime }}$ sends the tensor of the two unit vertex classes to the unit product vertex class. AC is inherited from [F2] for arbitrary-rank cycle retractions and simultaneous degreewise choices; composing the maps adds no choices.