I've seen the fact that the loopspace $\Omega K(G,n)$ is homotopy equivalent to $K(G,n-1)$ mentioned in some places, but I have no idea why. Can anyone offer a good explanation? Also, what happens when $n=1$? Does that just mean the loopspace is a discrete space with $|G|$ points?



Loopspace of an Eilenberg Maclane space K(G,n)
In general,the map $P(X,x\_0)\to X$ from the space of based paths in $X$ is a fibration with fiber $\Omega(X,x\_0)$. Since $P(X,x\_0)$ is contractible, by shrinking paths back toward the base point $x\_0$, the homotopy long exact sequence of the fibration shows that $\pi\_k(X,x\_0)\cong \pi\_{k-1}(\Omega(X,x\_0))$. The result follows. And, yes, the loop space on a $K(G,1)$ is discrete, in one-to-one correspondence with $G$.


