$R^2-\{x_1,x_2,\dots,x_n\}$ does not have the structure of topological group

Question

Let $n>1$. I need to show that the space $X=\mathbb{R}^2-\{x_1,x_2,\dots,x_n\}$ does not have the structure of topological group.

This is an exercise about the Van Kampen theorem. Certainly, we should prove it by contradiction, but I do not know how to get this contradiction.

Answer 1

Fundamental group of a topological group is abelian always. And it's easy to show the given space has non-abelian fundamental group.

Proof of the statement in bold:

Let $a$ and $b$ be two loops in a topological group $(G,\bullet )$ starting at the identity element $e$. We need to show $ a\ast b \simeq b\ast a$, where "$\ast$" is the fundamental group operation.

Now for each $t,s\in [0,1]$, define

$F_t(s)=a(st)\ast(a(t)\bullet b(s))\ast \bar a(st)$

Clearly {$F_t$} gives the homotopy between $b$ and $a\ast b \ast \bar a$.

( We can assume the topological group path connected )