TITLE: Is there a relationship between a quotient group of the fundamental group of X and the fundamental group of a quotient topology of X?
QUESTION [17 upvotes]: Let ($X$, $x_0$) be a topological space with a base point, and denote the fundamental group of $X$ as $\pi_1(X)$. Let $N$ be a normal subgroup of $\pi_1(X)$.
Does there necessarily exist an equivalence relation $\thicksim$ on $X$ such that $\pi_1(X/\thicksim)$ is isomorphic to $\pi_1(X)/N$?

REPLY [2 votes]: Not exactly an answer to your question, but it might be interesting anyway. Suppose your space $X$ is nice enough (path connected, locally path connected, semi-locally simply connected). Then under the Galois correspondence, the normal subgroup $N \triangleleft \pi_1(X)$ corresponds to a path-connected cover $p : Y \to X$ such that $\pi_1(Y) = N$ and $p_* : \pi_1(Y) \to \pi_1(X)$ is the inclusion.
Consider the mapping cylinder $M_p = (Y \times [0,1]) \cup_p X$. The covering $p : Y \to X$ factors as $Y \xrightarrow{\tilde{p}} M_p \xrightarrow{r} X$, where $r$ is a homotopy equivalence (a deformation retract). Therefore you have $\pi_1(M_p) = \pi_1(X)$ and $\tilde{p}_*$ induces the same map as $p_*$, i.e. the inclusion.
Finally, you have the long exact sequence of relative homotopy groups:
$$\dots \to \pi_1(Y) \to \pi_1(M_p) \to \pi_1(M_p, Y) \to \pi_0(Y) \to \pi_0(X) \to \dots$$
and since $X$ and $Y$ are connected, $\pi_0(Y) \to \pi_0(X)$ is an isomorphism. Thus
$$\pi_1(M_p, Y) = \pi_1(M_p) / \pi_1(Y) = \pi_1(X)/N.$$
Finally if your spaces are super-nice (CW-complexes), then there is a surjection $\pi_1(M_p, Y) \to \pi_1(M_p / Y)$, where $M_p / Y$ is the mapping cone of $p$. So you can view $\pi_1(X)/Y$ as a quotient of the fundamental group of a quotient of a space which deformation retracts onto $X$. I don't know (think) that you can say something better than that.