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Let $$E =\{(x,0) \in \mathbb{R}^2 \colon x \in \mathbb{Q} \}$$ $$F = \{(x,0) \in \mathbb{R}^2 \colon x \in \mathbb{R} \setminus \mathbb{Q}\}$$ compute the fundamental group of $\mathbb R^2\setminus E$ and $\mathbb R^2\setminus F$. How can I start?

(I don't know why the { symbols don't appear)

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  • $\begingroup$ I voted to close since it looked a bit like homework. However, the more I think about the question, it seems non-trivial and interesting. Anyhow, I cannot undo by vote but still want this question to stay open. $\endgroup$
    – Andreas Thom
    Commented Nov 20, 2010 at 20:55
  • $\begingroup$ If we enumerate the rationals, this space can be expressed as an inverse limit of spaces we know about. But topological functors seem to behave badly with inverse limits. Can we gain any information from this description of the firsts space? (I'm thinking about homology now; I've given up on the fundamental group.) $\endgroup$
    – Dylan Wilson
    Commented Nov 20, 2010 at 21:07
  • $\begingroup$ @Dylan One thought might be to calculate the shape fundamental group using that inverse limit, another would be using a van Kampen theorem -- I have not given up on the fundamental group. The question also reminds me of the calculations of the fundamental group of the Hawaiian earing (I used to know that but cannot be sure of the result now. I think the paper I knew was by H. B. Griffiths.) $\endgroup$
    – Tim Porter
    Commented Nov 20, 2010 at 21:18
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    $\begingroup$ I cast a vote for the question to remain open. $\endgroup$
    – Pete L. Clark
    Commented Nov 21, 2010 at 14:07

3 Answers 3

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When you want to compute the fundamental group of a wild space very often the thing to do is identify it as the subgroup of an inverse limit of simpler fundamental groups (often the first shape group). A result of Fischer and Zastrow says that if $X\subseteq \mathbb{R}^{2}$, then the canonical homomorphism of $\pi_1(X,x)$ into the shape group $\check{\pi}_{1}(X,x)$ is injective for any $x\in X$. Of course, this homomorphism is not always injective (even for 2-dimensional compacta) but for a subset of the plane like you have this approach should work. This is, for instance, how you compute the fundamental group of the Hawaiian earring. I would begin by looking for some simple approximating spaces (probably with free fundamental groups) with projection maps and figuring out which elements of the inverse limit of the fundamental groups of these spaces are represented by loops.

Here is the paper I mentiond:

Fischer, Zastrow, The fundamental groups of subsets of closed surfaces inject into their first shape groups. Algebraic and Geometric Topology. Volume 5 (2005) 1655–1676.

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These groups are rather ugly.
I don't know what one might possibly mean by "computing" them.

Here's an example of a path that you could use to construct an element in $\pi_1(\mathbb R^2\setminus E)$: the graph of the function $y=x\sin(x)$ (appropriately shifted so that it doesn't cross the x-axis at the origin). You can construct elements of $\pi_1(\mathbb R^2\setminus F)$ with similarly pathological behaviour.

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One point to start is to look at recent work by R. Diestel and P. Sprüssel

The fundamental group of a locally finite graph with ends, to appear in Advances in Mathematics

Your examples are probably not covered by their results, but maybe you can use the methods.

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