Eilenberg--Mac Lane spaces represent singular cohomology

By [F1], identify $H_n(K)$ with $A$. In the UCT exact sequence, the group to the left of evaluation vanishes for the reasons in [F2]. Therefore evaluation is an isomorphism, and there is a unique class $\iota$ satisfying

For a based map $f:X\to K$, let $\mathrm{\Psi }(f)$ be the primary difference class from $f$ to the constant map, relative to $x_0$. All lower obstructions vanish by [F4], so the required prior-stage homotopy exists. The difference theorem makes $\mathrm{\Psi }(f)$ independent of that homotopy and of the cellular choices and makes it invariant under based homotopy.

For the classes defined in Step 1.2, concatenate a lower-stage homotopy from $f$ to the constant map with the reverse of one from $g$ to the constant map. On each oriented $n$-cell, the resulting difference sphere splits along its equator into the sphere for $f$ and the oppositely oriented sphere for $g$. Hence, first as cochains and then as classes,

To realize values of $\mathrm{\Psi }$ from Step 1.2, let $c$ be a relative cellular $n$-cocycle representing an arbitrary class in ${\tilde{H}}^n(X;A)$ under [F3]. Collapse $X^{n-1}$ and, on the sphere belonging to each relative $n$-cell $e$, choose a based map to $K$ representing $c(e)\in A={\pi }_n(K)$. The CW wedge mapping property gives a map on $X^n\mathrm{/}{X}^{n-1}$. Its obstruction on an $(n+1)$-cell is exactly $(\delta c)(e^{n+1})=0$, so choose nullhomotopies and extend it over $X^{n+1}\mathrm{/}{X}^{n-1}$.

The construction of $\mathrm{\Psi }$ in Step 1.2 is natural for a cellular based map: its value on a source cell is obtained by evaluating the target cochain on the induced cellular chain. Cellular approximation and [F3] therefore give $\mathrm{\Psi }(fu)=u^{\ast }\mathrm{\Psi }(f)$ for every based CW map $u$.

Extend the map begun in Step 2.2: every later attaching obstruction lies in ${\pi }_r(K)=0$ for $r>n$. Inductively choose fillers and glue them to a based map $f:X\to K$, constant on $X^{n-1}$. By construction, its difference cochain from the constant map is $c$, so $\mathrm{\Psi }(f)=[c]$. Thus $\mathrm{\Psi }:[X,K{]}_{\ast }\to {\tilde{H}}^n(X;A)$ is surjective.

If $\mathrm{\Psi }(f)=\mathrm{\Psi }(g)$, Step 2.1 gives $[d(f,g)]=0$. By [F5], $f$ and $g$ are homotopic rel $x_0$ through $X^n$. Every obstruction to extending this homotopy across higher prism cells has coefficient ${\pi }_r(K)$ with $r>n$ and hence vanishes by [F4]. Induction and [A1] give a based homotopy on all of $X$. Thus $\mathrm{\Psi }$ is injective.

Apply the natural transformation of Step 2.3 to the identity of $K$. Choose its lower-skeleton homotopy to the constant map. On a relative Hurewicz $n$-cell generator, the difference sphere is the characteristic sphere on one hemisphere and constant on the other, so its class is the same element $u\in {\pi }_n(K)$. Consequently

Steps 3.1--3.3 prove the displayed natural bijection. The long exact sequence of $(X,\{x_0\})$ identifies relative and ordinary cohomology in every positive degree: in degree one the map $H^0(X;A)\to H^0(\{x_0\};A)$ is surjective, and in higher degrees the point groups on both sides vanish. This proves the final convention. The point, empty relative cell sets, $A=0$, and disconnected $X$ are covered componentwise. All arbitrary simultaneous choices occur only in Steps 2.2--3.2 and are covered by [A1]. $\mathrm{\square }$