Eilenberg-MacLane spaces in A-homotopy theory

In this paper we study a discrete homotopy theory for undirected graphs. More precisely, we consider $A$-homotopy groups, first defined in the work of Barcelo, Kramer, Laubenbacher and Weaver. For every degree $n$ and every group $G$ (Abelian if $n\ge 2$), we construct a graph whose $n$-th $A$-homotopy group is isomorphic to $G$ and whose other $A$-homotopy groups all vanish. Using these, we show that for every infinite sequence of groups $G_1,G_2,\dots$, with $G_i$ Abelian for $i\ge 2$, there is a graph $X$ such that the $i$-th $A$-homotopy group of $X$ is isomorphic to $G_i$ for all $i$. Along the way, we identify relative $A$-homotopy groups with relative classical homotopy groups, and prove a discrete version of the Blakers-Massey theorem. We then construct partial Postnikov systems and (full) Whitehead towers.