Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

Question

Let $Y:=\mathbb{R}^2-\{(0,1),(1,0),(-1,0)\}$. Calculate $\pi_1(Y,y_0)$, where $y_0=(0,0)$.

I think that this space is the free product $\mathbb{Z}*\mathbb{Z}*\mathbb{Z}$, but I do not know how to demonstrate this formally, could someone help me please? Thank you very much.

Answer 1

It is $\Bbb Z*\Bbb Z*\Bbb Z$. The plane with $n$ points removed has fundamental group the free group on $n$ generators. Reason: it's homotopic to a bouquet of $n$ circles.