Eilenberg-MacLane Spaces $K(G,n)$: Construction!

Question

I'm looking for an easy construction of $K(G,n)$, Eilenberg-MacLane spaces.

I know I can use the Postnikov Towers for the upper part $\pi_i(X)=0$ for $i > n$.

For the lower part $\pi_i(X)=0$ for $i < n$ and $\pi_n(X)=G$, I can use Cellular Approximation Theorem + Excision Theorem. Or Hurewicz Theorem + Moore Spaces.

I'm wondering if there is an easier way which avoids a so long path since I only have 1 hour in order to show the whole costruction only given the definition of $\pi_n$ with simple properties and the long exact relative sequence for a pair $(X,A)$, and maybe the Whitehead Theorem.

Thanks!

Answer 1

There exist constructions of the classifying space $BG$ of a topological group $G$ such that if $G$ is a topological abelian group, then so is $BG$ (e.g. due to Segal). Then $K(G, n)$ is $B^n G$.

If you know nice constructions of Moore spaces $M(G, n)$, then by the Dold-Thom theorem you can take the infinite symmetric product of these. For example, $K(\mathbb{Z}, n)$ is the infinite symmetric product of $S^n$.