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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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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. 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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\\{and\\}is to use backticks around all math. The problem is that the SE software is set to process everything with Markdown — actually, this is not a problem at all, since it allows you to italicize with underscores_or asterisks*. But Markdown thinks that{is a special character, and so interprets\{as an escape, and so strips off the\before sending the code to the browser. Then MathJax runs in the browser, and only sees{when you typed\{. Usually\\{works; sometimes you need\\\{; better is to protect everything. – Theo Johnson-Freyd Nov 20 2010 at 22:53