There exists a CW complex $K$ that is an Eilenberg-Maclane Space $K(\mathbb{Z}/2, n)$ with the following skeleton (for $n$ big enough, say $\ge 3$):
- One $e^0$.
- One $e^n$, whose boundary is collapsed to the basepoint. So $K^{(n)}=S^n$.
- One $e^{n+1}$, with $\partial e^{n+1}\to K^{(n)}$ of degree 2. This gives $\pi_n(K)=\mathbb{Z}/2$. Via homotopy long exact sequence, one gets $\pi_{n+1}(K^{(n+1)})=\mathbb{Z}/2$, via the Hopf map $S^{n+1}\to S^n \hookrightarrow K^{(n+1)}$, giving $f:S^{n+1}\to K^{(n+1)}$.
- One $e^{n+2}$ with $\partial e^{n+2}\overset{f}\to K^{(n+1)}$, to kill $\pi_{n+1}(K)=0$.
Unfortunately, when I tried to compute $\pi_{n+2}(K^{(n+2)})$, I needed $\pi_{n+2}(K^{(n+1)})$, which is either the Klein group or $\mathbb{Z}/4$. In general, it seems like proceeding to give an explicit CW structure for $K^{(n+3)}$ and $K^{(n+4)}$ would be very hard. Does anyone know if this CW structure has been studied? Thank you!