TITLE: is the group of rational numbers the fundamental group of some space?
QUESTION [27 upvotes]: Which path connected space has fundamental group isomorphic to the group of rationals? More generally, is every group the fundamental group of a space?

REPLY [21 votes]: Every group is the fundamental group of a space; a relatively easy choice of such a space is the presentation complex associated to a presentation. First, every group $G$ has a presentation with some generators $g_i$ indexed by some set $I$ and some relations $r_j$ indexed by some set $J$. Let $X$ be the wedge of $|I|$ circles; by Seifert-van Kampen we know that $\pi_1(X) \cong F_{|I|}$. 
Now we will add some $2$-cells corresponding to the relations. First, note that every relation $r_j$ determines a homotopy class of paths in $X$, hence a subspace isomorphic to $S^1$ of $X$. Associated to such a subspace is an attaching map $b_j : S^1 \to B^2$, and we can form the adjunction space $X \cup_{b_j} B^2$ in order to attach the appropriate $2$-cell and kill the relation $r_j$ (proven by a second application of Seifert-van Kampen). Doing this for all relations gives the appropriate space.