The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$.

There is also a generalization of the van Kampen theorem that holds for the fundamental groupoid of a space $X$, where in this case it states precisely that the fundamental groupoid functor, $\Pi$, preserves certain colimits in $Top$, namely, those that arise from "nice" open coverings of $X$.

The "groupoid" version of the van Kampen theorem seems to me more conceptual and more elegant than the classical version. Also, the groupoid version allows one to prove the classical version in a more or less easier way.

Although, apart from the conceptual advantages of the groupoid version of van Kampen theorem, I would like to know if we are able to do any interesting calculations using the fundamental groupoid version of the van Kampen. In fact, explicitly describing a groupoid as a colimit of "simpler" groupoids is something that it is not clear at all for me. I would like to know some concrete cases where is it possible to describe the fundamental groupoid of a space, using this generalization form of the van Kampen theorem, and, if possible, to calculate the fundamental group directly from our calculation of $\Pi(X).$

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    Have you searched in this site (and in math.stackexchange.com) for relevant questions and answers? I am sure this has been discussed before —in particular, by Ronnie Brown. – Mariano Suárez-Álvarez Jul 12 '14 at 10:43
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    The fundamental groupoid functor preserves all homotopy colimits, and that's very nice. – Fernando Muro Jul 13 '14 at 10:33
  • This appears to be a duplicate question: mathoverflow.net/questions/40945/… – Ryan Budney Oct 17 '16 at 17:39
  • See also mathoverflow.net/questions/220561/… for an opinion of Grothendieck. Is it possible to construct examples where a groupoid $G$ is "computable" but some of its vertex groups are not computable? – Ronnie Brown Aug 24 at 10:49
up vote 22 down vote accepted

I confess, in 1965 or so, I first thought that the version of the SvKT (Seifert-van Kampen Theorem) for the fundamental groupoid enabled us to get rid of base points. But then I wanted to calculate the fundamental group of the circle, and gradually realised that we needed $\pi_1(X,A)$, the fundamental groupoid on a set of $A$ of base points chosen according to the geometry. Here is an example of the kind of situation which the standard texts do not give:

fog

and for which covering space methods are not ideal. In fact one needs combinatorics and combinatorial group(oid) theory to calculate individual $\pi_1(X,a)$ from $\pi_1(X,A)$. See my book Topology and Groupoids and also the classic 1971 book (downloadable) by Philip Higgins, Categories and Groupoids.

Groupoids model homotopy 1-types. So one first determines the 1-type before calculating an individual fundamental group.

This idea is nicely modelled in higher dimensions: one can calculate some 2-types and higher types as "big" algebraic objects inside which are the homotopy groups which one might want. The methods are explained more in a talk given in Paris on June 5, 2014, at the IHP, available on my preprint page.

As Mariano points out, this has been discussed elsewhere on stackexchange and mathoverflow.

July 14: The most general version of the SvKT is given in [41] (downloadable) on my publication list,

R. Brown and A. Razak, `A van Kampen theorem for unions of non-connected spaces'', Archiv. Math. 42 (1984) 85-88.

in the form of a coequaliser statement when given an open cover and a set $A$ of base points which meets each path component of each 1-,2-, and 3-fold intersections of the sets of the cover. The style of proof goes back to Crowell's original version, and has the advantage of generalising to higher dimensions. For example, in

[32] R. Brown and P.J. Higgins, ``Colimit theorems for relative homotopy groups'', J. Pure Appl. Algebra 22 (1981) 11-41.

The retraction from the version for the full fundamental groupoid, $A=X$, to this version is quite difficult to manage and is done in May's "Concise..." book, without the refinement on the conditions, but I feel is the wrong way to go, though it is quite elegant for the pushout version in dimension 1.

July 17, 2014: Actually I have missed out three reasons why one is interested in the fundamental groupoid $\pi_1 X$.

  1. The notion of fibration of groupoids is relevant to topology particularly in construction of operations of groupoids on homotopy sets, and exact sequences. If $p: E \to B$ is a fibration of spaces then $\pi_1 p: \pi_1 E \to \pi_1 B$ is a fibration of groupoids. This is exploited in Chapter 7 of Topology and Groupoids. See for example arXiv:1207.6404 for other current uses of the algebra of groupoids, and in particular fibrations.

  2. Similarly if $p: E \to B$ is a covering map of spaces then on applying $\pi_1$ we get a covering morphism of groupoids. Thus a map is modelled by a morphism, and this often makes the theory easier to follow (IMHO!), particularly with regard to questions of lifting maps. See Chapter 10 of T&G.

  3. If $G$ is a (discrete) group acting on a space $X$ then it also acts on the fundamental groupoid $\pi_1 X$. So we have not only orbit spaces $X/G$ but also orbit groupoids $(\pi_1 X)/\!/G$. There is a canonical morphism $(\pi_1 X)/\!/G \to \pi_1(X/G)$ and there are useful conditions which ensure this is an isomorphism, e.g. $X$ is Hausdorff, has a universal cover, and the action is properly discontinuous. See Chapter 11 of T&G. One example given there is the cyclic group $Z_2$ acting on $X \times X$ whose orbit space is the symmetric square of $X$. Under useful conditions, its fundamental group is that of $X$ made abelian. It would be good to see lots more examples.

Aug 17, 2016: I can now refer to my answer to my own mathoverflow question on the relation of the van Kampen Theorem to the notion of descent.

Oct 17, 2016: I should add that a whole area of research into the use of strict higher groupoids in algebraic topology, one part of which is described in the book Nonabelian Algebraic Topology, (EMS 2011, 703 pages), arose out of seeking generalisations to higher dimensions of the use of the fundamental groupoid.

December 23, 2016 It may be useful to point out that a new volume by Bourbaki "Topologie alg\'ebrique" Ch 1-4, (Springer) 2016, uses the fundamental groupoid extensively, and relates its use to descent theory. It does have results on orbit spaces, but no example of applications. It does not use the fundamental groupoid on a set of base points.

I'm very fond of Ronnie Brown's proof of the Jordan curve theorem using the groupoid version of the van Kampen theorem. I find it easier to understand and reproduce than other comparably elementary proofs, such as Munkres' argument using covering spaces (which can be found in his textbook Topology: a first course). I guess this application isn't really a calculation, as requested, but I hope you like it.

(I guess I should mention that a small gap in the proof is filled in in this short note.)

  • I really appreciate your answer. Thanks! – Jorge António Jul 15 '14 at 19:06
  • Of course, one of the things that's going on here is that covering spaces are very closely related to groupoids: the fundamental group lifts to a groupoid of paths between the preimages of the base point. Probably (though I haven't checked) the groupoid and covering space versions are more or less the same proof from different points of view. – HJRW Oct 30 at 13:23

The simplest natural example is to use the van Kampen theorem (in Ronnie Brown's version) to work out the fundamental group(oid) of the circle. The details are in his book Topology and Groupoids (http://groupoids.org.uk/topgpds.html). That source also discusses colimits of groupoids in this context. If you look around on Ronnie's home pages (http://groupoids.org.uk) you can find discussion of many of the points that you raise.

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