Existence and homotopy uniqueness of Eilenberg--Mac Lane spaces

For $G$, begin with a wedge $W$ of one oriented circle $x_g$ for every $g\in G$. Attach a $2$-cell along the word $x_g{x}_h{x}_{gh}^{-1}$ for every ordered pair $(g,h)$. By [F1], the resulting complex $Q_2$ has presentation

Now let $n\ge 2$. Put $W={\bigvee }_{a\in A}{S}_a^n$. By [F2], ${\pi }_n(W)={\mathbb{Z}}^{(A)}$. Let $ϵ:{\mathbb{Z}}^{(A)}\to A$ send the basis vector $e_a$ to $a$. Choose a sphere representative for every element of $\ker ϵ$ and attach an $(n+1)$-cell along it, obtaining $Q_{n+1}$. The pair $(Q_{n+1},W)$ is $n$-connected and $W$ is simply connected. Relative Hurewicz identifies its relative ${\pi }_{n+1}$ with the free relative cell group, and the boundary map sends each cell generator to its attaching class. Exactness therefore gives

Starting from the degree-one presentation in Step 1.1, inductively choose one based map $S^r\to Q_r$ representing every element of ${\pi }_r(Q_r)$ and attach an $(r+1)$-cell along each, for $r\ge 2$. The relative exact sequence makes ${\pi }_r(Q_r)\to {\pi }_r(Q_{r+1})$ zero and surjective, hence ${\pi }_r(Q_{r+1})=0$, while [F3] preserves all lower groups. Put $Q={\bigcup }_{r\ge 2}{Q}_r$. For fixed $i>1$, later cells do not recreate ${\pi }_i$. By [F4], every representative and nullhomotopy in $Q$ occurs at a finite stage; consequently ${\pi }_1(Q)=G$ and ${\pi }_i(Q)=0$ for $i>1$. Thus $Q$ is a $K(G,1)$.

Starting from the degree-$n$ complex in Step 1.2, attach one $(r+1)$-cell along a representative of every element of ${\pi }_r$ at the current stage, beginning with $r=n+1$. The argument of Step 2.1, now preserving ${\pi }_n=A$, kills each higher group successively. The increasing union $Q$ has ${\pi }_n(Q)=A$ and every other positive homotopy group zero by [F4]. This is a $K(A,n)$.

Let $K$ be any other degree-one model with the same identified group. From the completed model in Step 2.1, map each circle $x_g$ to a based loop representing the corresponding $g\in {\pi }_1(K)$. Each multiplication relator maps to a nullhomotopic loop, so choose fillings of the $2$-cells. Every higher attaching sphere maps trivially because ${\pi }_r(K)=0$ for $r>1$, and induction extends the map to $u:Q\to K$. It induces the prescribed isomorphism on ${\pi }_1$.

For a degree-$n$ model $K$, begin with the completed model in Step 3.1 and map the sphere indexed by $a$ to a representative of the corresponding element of ${\pi }_n(K)$. Every attaching map indexed by $\ker ϵ$ becomes nullhomotopic, so the map extends over the $(n+1)$-cells. All later attaching maps extend because the corresponding higher homotopy groups of $K$ vanish. The resulting $u:Q\to K$ induces the prescribed isomorphism on ${\pi }_n$.

In either case, $Q$ and $K$ are connected, $u$ is an isomorphism on their sole possibly nonzero positive homotopy group, and all their other positive homotopy groups vanish. Thus $u$ is a weak equivalence. By [F5], it is a homotopy equivalence. Applying Steps 3.2 and 4.1 to two models gives a zigzag of homotopy equivalences through $Q$; choosing a homotopy inverse for one leg gives a homotopy equivalence between the models. Its induced group map is the prescribed identification, with the basepoint track supplying the standard conjugacy transport when $n=1$.

The construction also covers the trivial group. In that case the resulting connected CW complex has every positive homotopy group zero, and its map to a point is a weak equivalence, hence a homotopy equivalence by [F5]. No countability, finite generation, or finite-dimensionality has been assumed. Every infinite selection is accounted for by [A1], while the passage to the union uses the individual compact-support statement [F4], not an unproved interchange of homotopy groups with an arbitrary colimit. $\mathrm{\square }$