UCT and Kunneth collapse retains an extension problem

With only resolution columns zero and one, the decreasing UCT filtration is $0\subset F^1{H}^n\subset H^n$, giving $0\to E_{\mathrm{\infty }}^{1,n-1}\to H^n\to E_{\mathrm{\infty }}^{0,n}\to 0$. The increasing Künneth filtration instead gives $0\to E_{0,n}^{\mathrm{\infty }}\to H_n\to E_{1,n-1}^{\mathrm{\infty }}\to 0$. Collapse identifies these graded terms with the second page but selects no section of either quotient. These are kernel/quotient constructions requiring no choice.

The obstruction is concrete: $0\to 2\mathbb{Z}\mathrm{/}4\to \mathbb{Z}\mathrm{/}4\to \mathbb{Z}\mathrm{/}2\to 0$ has both graded pieces $\mathbb{Z}\mathrm{/}2$, but cannot split. A lift of the quotient generator is $1$ or $3$ modulo four, and neither is killed by two. F3 realizes precisely this ambiguity in a collapsed filtered complex. This is a general collapse example, not a claim that this nonsplit extension is realized by free-PID UCT.

In the separate PID UCT setting of F4, AC makes $B_{n-1}C$ projective and permits a section of $C_n\to B_{n-1}C$, hence a retraction $r:C_n\to Z_{n}C$. For $\varphi :H_{n}C\to M$, the cochain $\varphi \pi r$ is a cocycle: on $d_{n+1}{C}_{n+1}=B_{n}C$, $r$ is the identity and $\pi$ kills it. Its cohomology class evaluates to $\varphi$, and dependence on $\varphi$ is additive. Thus this gives a section of the UCT quotient. F5 supplies the required choices when sections in all degrees are wanted. The Künneth splitting is supplied by the completed construction in F2.

For UCT nonnaturality use $C_1=\mathbb{Z}a\oplus \mathbb{Z}b$, $C_0=\mathbb{Z}c$, $da=2c,db=0$, and $M=\mathbb{Z}\mathrm{/}2$. Then $H_{0}C=\mathbb{Z}\mathrm{/}2$, $H_{1}C=\mathbb{Z}$, and $H^1\mathrm{Hom}(C,M)=M^2$ because the incoming coboundary is $(2m,0)=0$. Evaluation is $(x,y)\mapsto y$ and its kernel is the first coordinate. The chain automorphism $a\mapsto a+b$, fixing $b,c$, fixes both homology groups but acts on $M^2$ by $(x,y)\mapsto (x+y,y)$. No lift $(x,1)$ of $1$ is fixed, so no section can be natural in $C$. F2 gives the corresponding tensor shear obstruction for Künneth. Zero filtration pieces may remove an individual extension problem, but cannot turn this counterexample into a natural splitting theorem.