TITLE: How to construct $K(\mathbb{Z}/5\mathbb{Z},1)$?
QUESTION [7 upvotes]: From the Wikipedia article on Eilenberg-MacLane spaces:

A $K(G, n)$ can be constructed stage-by-stage, as a CW complex, starting
  with a wedge of n-spheres, one for each generator of the group $G$, and
  adding cells in (possibly infinite number of) higher dimensions so as
  to kill all extra homotopy.

I have a vague intuition of how this would work—for, say, $\mathbb{Z}/5\mathbb{Z}$—but I haven't been able to locate a step-by-step description of such a construction. I know the fact that this example should require adding cells in every higher dimension. I'd appreciate if you could explain this to me or if you could point me to such an explanation. All the better if it talks through what's happening with the geometry—how exactly does twisty-gluing that first disk into the single loop create a sphere(?) that then needs to get killed? And how does killing this hole lead to homotopy in the next higher dimension, etc.?

REPLY [5 votes]: $K(Z_5,1)$ is infinite Lens space.So it is actually $S^{\infty}/Z_5$.
Another method: Since $\pi_1(K(Z_5,1)) =Z_5=<a|a^5 =1>$.Then we can construct $X$ as follows:
Take one circle.Attach one $2$-cell with attaching map $z \rightarrow z^5$.Now attached $n+1$ -cell to cancel $\pi_n$ for all $n>1$.This $X$ will serve one model for $K(Z_5,1)$.