Every Group is a Fundamental Group

Question

I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and for any $x \epsilon X$ there is a fundamental group $\pi_1 (X,x)$. I have a question about the converse. If we consider any arbitrary group $G$, then does there exist a topological space $X$ and an element $x\epsilon X$ such that $\pi_1 (X,x)=G$?

Answer 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 $$\Bbb Z\times \Bbb Z\cong F(a,b)/\langle aba^{-1}b^{-1}\rangle.$$