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How can I calculate de fundamenta group of $\mathbb{R}^2$ without a finite number of points?

I know that the answer should be $S^1\vee S^1 \vee \cdots \vee S^1 $ where the product repeats as many as points removed.

Thanks for helps!

EQJ
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1 Answers1

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Do a proof by induction that the group is free of rank~$n$ where $n$ points have been removed. The induction step uses Van Kampen's theorem. Letting the points be $x_1,\ldots,x_n$, there exists a line $L$ disjoint from $\{x_1,\ldots,x_n\}$ such that each component of $\mathbb{R}^2-L$ contains at least one of the points. Renumber the points so that $x_1,\ldots,x_k$ are in one side and $x_{k+1},\ldots,x_n$ are on the other. By induction, the first side has fundamental group $F_k$ (the free group of rank $k$) and the other has fundamental group $F_{n-k}$. By Van Kampen's theorem, the fundamental group of the whole of $\mathbb{R}^n-\{x_1,\ldots,x_n\}$ is the free product of $F_k$ and $F_{n-k}$ which is $F_n$.

Lee Mosher
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    Wonderful! This proof avoid every type of complication (or intuition) as deformation retracts. Thanks! – EQJ Jun 18 '15 at 00:45
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    Take lines on either side of $L$, parallel to it, and very close. $U$ is a half plane bounded one of of those lines, $V$ a half plane bounded by the other. The intersection is a narrow strip centered on $L$, which is simply connected. – Lee Mosher Jun 18 '15 at 00:45
  • But when you invoke with induction hypothesis, that $\mathbb{R}^2 - {k-points}$ has fundamental group of a wedge of $k$ circles, we are really looking at half a plane with boundary (the line L), so is it technically correct to use the induction hypothesis? The induction hypothesis I assume would be that if $k \leq n$ then the fundamental group of the plane minus $k$ points the same as a wedge of $k$ circles. –  Jun 01 '19 at 18:28
  • No, the induction hypothesis is that my first sentence is true for $n$ replaced by any value $1 \le k \le n-1$. Deformation retractions have nothing to do with this particular method. – Lee Mosher Jun 01 '19 at 18:32
  • @LeeMosher I've got two questions : What could it be a way to define "explicitly" such $L$? And second, how do you prove that those are open set in order to apply van Kampen correctly ? – jacopoburelli Sep 08 '20 at 20:45