Let $X:=\mathbb R^2\setminus\{x_0,x_1\}$, where $x_0,x_1\in\mathbb R^2\setminus\{0,0\}$. Compute the fundamental Group of $\pi_1(X,(0,0))$

Question

Let $X:=\mathbb R^2\setminus\{x_0,x_1\}$, where $x_0,x_1\in\mathbb R^2\setminus\{0,0\}$. Compute the fundamental Group of $\pi_1(X,(0,0))$

What does mean ''compute'' ? i can only draw it.

or can we define a deformation retract in this case ?

Answer 1

Note that the space described can be deformation retracted on to a subspace which is a figure eight (wedge of two circles), containing $(0,0)$ at the wedge point. You should prove this statement (hint, the two circles making up the figure eight should each be loops around one of the removed points).

You should have met a theorem which says that, for sufficiently nice spaces, if $\pi_1(X,x)=G$ and $\pi_1(X',x')=G'$, then $$\pi_1(X\vee X',[x])\cong G\ast G'$$ which can be proved easily using Van Kampen's theorem.