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!

  • $\begingroup$ If it helps, note that the original space is homeomorphic to $\mathbb C$ with the roots of $x^n-1$ removed. $\endgroup$ – 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.


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