Recall that an Eilenberg-Maclane space $K(G, n)$ is characterized by $\pi_i(K(G,n)) = G$ if $i=n$ and is trivial otherwise. (Of course $G$ should be abelian if $n>1$.)

Let $G$ be a finite abelian group.

Below I describe cell complexes $X_1$ and $X_2$ with $\pi_2(X_i) = G$ and $\pi_0(X_i)$ and $\pi_1(X_i)$ both trivial. By standard results it is possible to add 4-cells to $X_i$ to kill off $\pi_3$, then add 5-cells to kill off $\pi_4$, and so on.

My questions:

(1.i) Does there exist in the literature an explicit description of the 4- and 5-cells one would need to add to $X_i$ in order to turn it into a $K(G,2)$? (I'm only interested in dimensions 4 and 5, not higher.)

(2) More generally, are there explicit descriptions of $K(G,2)$ in the literature? (I'm already aware of making $K(G, 1)$ into a group and then applying the bar construction.)

**Definition of $X_1$:** A single 0-cell. A 2-cell $c_g$ for each element $g\in G$. A 3-cell $d_{g,h}$ for each $(g,h)\in G\times G$, with $\partial d_{g,h} = c_g + c_h - c_{gh}$.

(This starts out similarly to a standard construction of $K(G, 1)$, but the higher dimensional cells will necessarily be more complicated. Obvious candidates for the boundaries of 4-cells would include $d_{g,h} - d_{fg,h} + d_{f,gh} - d_{f,g}$ for all $(f,g,h)\in G\times G\times G$, and also Hopf maps to the 2-cells $c_g$ for each $g$, and also $d_{g,h} - d_{h,g} + x$, where $x$ is a map to $c_h\cup c_h$ which exhibits the commutativity of $\pi_2(c_g\cup c_h)$.)

**Definition of $X_2$:** Let $G = \mathbb Z/k_1 \times\cdots\times \mathbb Z/k_m$, a prodict of cyclic groups. $X_2$ has $m$ 2-cells $e_1,\ldots, e_m$ and $m$ 3-cells $f_1,\ldots, f_m$, with $\partial f_i = k_i\cdot e_i$.