# Use Van Kampen theorem to find the fundamental group

I'm got stuck in calculating fundamental group of the plane ($$\mathbb{R}^2$$) minus finitely many points ($$n$$ points).

I think the space is homotopic the wedge sum of n - circles ($$S^1$$). And its fundamental group is isomorphic to $$\pi_1(S^1, p_1)_*...\pi_1(S^1, p_n)_*$$, but I can't write exactly what is the homotopy is.

Someone can help me!!! Thank you!

• If it helps, note that the original space is homeomorphic to $\mathbb C$ with the roots of $x^n-1$ removed. – Rafay Ashary Dec 26 '18 at 5:50

You can prove the homotopy you are mentioning, which would be a bit tedious; or you can try the following : prove that your space is homeomorphic to $$\mathbb{R}^2$$ minus $$n$$ points on a straight line, say the $$x$$-axis.
Then proceed by induction with Van Kampen in the following way : take the leftmost point, and consider an open set containing only this point of the form $$\{(x,y)\in \mathbb{R}^2\mid x< a\}$$ . There is an $$a$$ that will work.
Then you can take an open set of the form $$\{(x,y)\in \mathbb{R}^2\mid x>b\}$$ that contain the rest of the points; the intersection of these two open sets will be a stripe which is contractible; and the two open sets will be homeomorphic to $$\mathbb{R}^2$$ minus one point, or $$n-1$$ points respectively, thus you can use Van Kampen, the $$n=1$$ case and induction to conclude.