After learning about the fundamental group, and proving that $\mathbb{R}^n$ minus any countable set is path-connected, I started wondering if the fundamental group of $\mathbb{R}^2-\mathbb{Q}^2$ is known. Does anyone know whether or not it is? Or how one might go about determining it?
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asked Jun 11, 2011 at 22:44
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3$\begingroup$ This math.stackexchange question might help. math.stackexchange.com/q/16948/1321 $\endgroup$– George LowtherCommented Jun 11, 2011 at 23:30
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$\begingroup$ An exercise in Hatcher's textbook asks to show that it is uncountable. $\endgroup$– Chris GerigCommented Jun 11, 2011 at 23:36
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$\begingroup$ What does it mean for a group to be "known"? Fundamental groups of subsets of the plane have a certain characterization (Eda 1998). But even finitely-presented groups have no classification, and this group isn't even finitely generated... $\endgroup$– Ryan BudneyCommented Jun 11, 2011 at 23:49
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$\begingroup$ Perhaps the simple question of 'what is this fundamental group?' is not a good fit for MO. But I think there is something to be learned from this example, namely, you don't really want the fundamental group in this case, but something else. This 'something else' is detailed in my answer. $\endgroup$– David Roberts ♦Commented Jun 12, 2011 at 0:03
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The space $X = \mathbb{R}^2 - \mathbb{Q}^2$ is not semilocally simply connected, and so in a sense the fundamental group is a poor measure of the homotopy 1-type of $X$. It is an exercise in Hatcher's book Algebraic Topology that this group is uncountable.
Here is an attempt to motivate a different view of the fundamental group: Consider the space $X_{n,m}$ which is the complement in $\mathbb{R}^2$ of all points with coordinates $(p/n,q/m)$ where $n,m$ are positive integers and $q,p$ are integers. This forms a cofiltered diagram $\mathbb{N}^2 \to Top$ whose limit is $X$, and where each space in the diagram is semilocally simply-connected. There is a map $X_{n,m} \to X_{n',m'}$ when $n'|n$ and $m'|m$. The fundamental group of $X_{n,m}$ is free on a countable number of generators (a generator is given by a small circle around each deleted point, and connect this circle to the basepoint, $(\pi,\pi)$, say, by a path), and the maps between them kill off certain generators. Already, you can see how complicated $\pi_1(X)$ is going to be, since it has a countable number of quotients, each of which is free on a countable number of generators (actually, each of them is isomorphic to the free group on generators $\mathbb{Z}\times\mathbb{Z}$).
This system of groups is known as a (strict) progroup, and can be thought of as a formal limit This is usually the 'right' algebraic object to consider when one has a badly behaved space such as this one. It is this object which 'controls' what the covering spaces of $X$ look like, as $X$ doesn't have a universal covering space (which if it did exist, would have fibre $\pi_1(X)$, and other connected covering spaces would be quotients of this one). This gets into the realm of shape theory (wikipedia, nLab), developed to consider spaces with bad local homotopy properties.
answered Jun 12, 2011 at 0:00
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3$\begingroup$ It is worth noting that this pro-group is not profinite so taking the limit is not a good thing to do. As David says you need to look at the pro-group as it is. No single (discrete) group will do the job for you. There are interesting old results on the fundamental group of the Hawaiian earring that discuss topologies on the group that help. I do not know if that would help here. (Why do you want to know? Is it just curiosity? I ask because the style of answer will be influenced by any intended use.) $\endgroup$ Commented Jun 12, 2011 at 9:37
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$\begingroup$ In what sense is this construction canonical? One could produce progroups using other diagrams with limit X; are these progroups always equivalent, assuming some hypotheses on the diagram? $\endgroup$ Commented Jun 12, 2011 at 17:25
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$\begingroup$ Is this approach at all related to David Bliss' "A Generalized Approach to the Fundamental Group"? Also, could you point me to a (preferably online) description of shape theory that someone with only minimal experience with category theory could follow? The wikipedia article doesn't give any definitions, and the shape theory definition for topological spaces is missing from the nLabs article. @Tim Porter: It's just curiosity. $\endgroup$ Commented Jun 12, 2011 at 21:27
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$\begingroup$ @Dan - it's not canonical at all (really just the first construction that came to mind) but my guess from what I know of shape theory is that this choice of 'resolution' of $X$ doesn't matter, up to some sort of weak equivalence in the so-called shape category. @Avi - actually there is a mistake in that paper in that the fundamental group is not always a topological group, only a quasi-topological group (multiplication is separately continuous in each variable). There would be such a structure floating around here, but as always, it depends on what you want to do with the fundamental group.. $\endgroup$– David Roberts ♦Commented Jun 12, 2011 at 23:43
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$\begingroup$ ... If you are interested in (ordinary) covering spaces, then the above is the right way to do it. There are also approaches using topoi and localic fundamental groups (work by Marta Bunge is appropriate here), fibre functors and Galois categories (due to Grothendieck, discussed in a nice paper by Eduardo Debuc) - which is close to what I mention above, but removes the choice of resolving $X$ by the $X_{n,m}$ - it considers all covering spaces. As far as shape theory goes, it's a slippery beast. The 'right' way to do it is how Tim and others have since developed it, and that is abstractly.... $\endgroup$– David Roberts ♦Commented Jun 12, 2011 at 23:50