TITLE: Example of a space that has fundamental group with a specific representation
QUESTION [6 upvotes]: I'm trying to solve a specific task of finding a space with $\pi_1(X) = <a,b,c | a^2cb, abc>$, but I'm interested if there is a way to think about this in general. I understand fairly intuitively the definition of a fundamental group and what a representation like this means algebraically. I'm completely lost when it comes to the geometry though and how one might go about constructing such a space. I'd be grateful if someone could do this or a similar example, or for any tips on the intuition behind such constructions.

REPLY [3 votes]: Bouquets of circles give you a supply of spaces whose fundamental groups are free groups with $1$ generator for each circle in the bouquet. E.g., in your example the bouquet of three circles $A$, $B$ and $C$ has fundamental group the free group $F = <a. b, c>$. A relation word in $F$ of length $n$ (e.g.,  $abc$ with $n=3$) corresponds to a continuous mapping from the edges of an $n$-gon (e.g., a triangle) into the bouquet (that starts at the common point, then goes once round $A$, once round $B$ and once round $C$). If you form a new space by stitching on an $n$-gon to the bouquet for each relation word using these mappings to identify the edges of the $n$-gon with circles in the bouquet, you get a space whose fundamental group is as desired.