Topological universal coefficient short exact sequence for cohomology

The absolute integral singular complex is nonnegative and free in every degree by [F2]. The ring $\mathbb{Z}$ is a PID. Applying [F1] gives the displayed short exact sequence, with its middle group identified as $H^n(X;G)$ by [F2]. On a cycle $c$, the right map sends $[\phi ]$ to $([c]\mapsto \phi (c))$, so it is precisely the Kronecker evaluation of [F4].

For the pair, the relative chain group is freely generated by those singular simplices of $X$ whose image is not contained in $A$: discard the coefficients on simplices in $A$ from a finite chain. Equality of the resulting complementary coefficients is exactly equality modulo chains in $A$. This basis identification is degreewise; its differential is the induced quotient differential, which need not preserve zero extensions. Thus [F3] supplies a nonnegative free complex whose Hom cohomology is $H^n(X,A;G)$. Apply [F1] to it. Evaluation on relative cycles descends by the same calculation in [F1], so no absolute cycle lift is required.

A continuous pair map $f:(X,A)\to (Y,B)$ sends a simplex in $A$ to a simplex in $B$, so the absolute chain map of [F4] descends to relative quotients and still commutes with boundaries. Its Hom pullback is exactly relative cochain precomposition. Hence chain naturality in [F1] proves the commuting short-exact-sequence diagram for pairs, as well as for absolute maps. Coefficient postcomposition is the coefficient map in both Hom complexes and in the displayed Ext cokernels, so coefficient naturality in [F1] gives the second type of commuting diagram. These assertions concern canonical evaluation and extension maps, independent of projections used to prove surjectivity.

At $n=0$, the left Ext group is zero by [F1] and evaluation is an isomorphism. For $A=\mathrm{\varnothing }$, step 1.2 reproduces step 1.1; for $A=X$, all relative chain groups and all terms of the sequence are zero. Empty $X$ and $G=0$ are included. On a point, the degree-zero evaluation sends a cochain value $g$ to the homomorphism $m\mapsto mg$. AC is inherited from [F1] for arbitrary-rank cycle/boundary freeness and projections, while the complementary simplex basis and the naturality maps require no additional choices. This proves every assertion.