Every Group is a Fundamental Group

Solution 1:

That statement is true. First, every free group $F$ is a the fundamental group of a bouquet of circles (having a common base point); each circle represents a generator. Now every group $G$ is a quotient of a free group $F$ by a subgroup (of $F$) generated by the "relations" $f_i\in F$.

Note that each $f_i$ is a word in $F$, that is, a product of the generators (and their inverses) of $F$. We then "kill" the relation $f_i$ by attaching a disk $D^2$ along the circles in $f_i$.

As a (very good) practice problem, try to show the fundamental group of the torus is $$\Bbb Z\times \Bbb Z\cong F(a,b)/\langle aba^{-1}b^{-1}\rangle.$$