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Hi, Recently I've been looking at the more general version of Van Kampen's theorem, or R. Browns version of it, for the fundamental groupoid. It mentions that if a space X is the union of the interiors of $X_1$ and $X_2$ , then:

alt text

is a pushout in the category of groupoids. In the category of groups, we have the concrete description that the pushout is just the free product with amalgation. Does something similar hold here? Is there any explicit description, like free product with amalgation?

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"push-out" and "free product with amalgamation" are pretty much synonymous, so it's not clear to me there's anything happening to generalize. –  Ryan Budney Sep 3 '11 at 22:13
    
I think Ryan is saying that the "free product with amalgamation" interpretation of a pushout of groups is just a general categorical fact: in any category with finite coproducts and coequalizers, the pushout of $A \rightarrow B$ along $A \rightarrow C$ is the coequalizer the two maps into the coproduct of $B$ and $C$. –  Dylan Wilson Sep 3 '11 at 22:28
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I would use "free product with amalgamation" only when the upper corner group injects into the other two groups. This is also the case when you have a really nice description of the result. Otherwise you only get a presentation (given presentation for the three groups) and we all know how difficult such can be to handle... –  Torsten Ekedahl Sep 5 '11 at 10:38

4 Answers 4

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I agree with the comments above: being a pushout is a categorical property. What is useful is to be able to compute explictly such pushouts and, as you say, free/amalgamated products do so in the category of groups.

In his paper Le théorème de Van Kampen (Cahiers de Topologie et Géométrie Différentielle Catégoriques, 33 no. 3 (1992), p. 237-251. Available on Numdam, http://www.numdam.org/item?id=CTGDC_1992__33_3_237_0), André Gramain gives (part of) an explicit recipe to compute the isotropy groups of a coequalizer of a pair $(\phi,\psi)$ of morphisms of groupoids. This recipe applies to your case by considering (as in van Kampen's theory) the disjoint sum of the groupoids $\pi_1(X_1,A)$ and $\pi_1(X_2,A)$ and the two morphisms from $\pi_1(X_0,A)$ to this disjoint union.

In SGA 1 (Revêtements étales et groupe fondamental, Exposé IX, §5), Grothendieck had given the same recipe for the fundamental group of schemes. However, his proof is more categorical and based on the correspondence between coverings and sets with action of the fundamental groups, and on descent theory for coverings.

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Andre Gramain's 1992 exposition of van Kampen's statement is in fact covered explicitly in "Topology and Groupoids", as it was in the 1988 version of that book; it was just an exercise in the 1968 edition.

The point of this type of exposition is to say that sometimes a group can be better explicitly described in terms of groupoids. A basic example is the group $\mathbb Z$ of integers! This is obtained from the groupoid $\mathcal I$ which has two objects 0,1 and exactly one arrow between them by identifying 0 and 1. This is rather analogous to the way the circle is obtained from the unit interval $[0,1]$ by identifying 0 and 1 !

Another aspect is that sometimes a groupoid is a better object to deal with than a group. For example, a homotopy colimit of a diagram of groups is really a groupoid. This is analogous in topology to taking double mapping cylinders rather than a pushout of maps of CW-complexes.

For more information on the book see http://pages.bangor.ac.uk/~mas010/topgpds.html

I should also say the Higgins' monograph has results on groups, for example a generalisation of Grusko's theorem, that has not been equalled by other methods.

Another aspect of Topology and groupoids is Chapter 11 on "Orbit spaces, orbit groupoids", which allows some computation of the fundamental groupoid and hence group of an orbit space.

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I don't fully agree: while it is obvious that van Kampen theorems for groupoids have been published long ago, notably in your book, I am not able to recognize there the formulae of which Gramain gives a preview (beyond the case of quotient by a groupp actions, as in your Chapter 11). But I would love to learn I'm wrong ! –  ACL Nov 23 '11 at 13:30
    
Another comment: I like to compute the fundamental group of the circle as follows. The circle is the interval with endpoints 0 and 1 identified. A covering of the circle is a covering of the interval together with an identification of the fibers at 0 and 1. A covering of the interval is trivial, so is just a set ; identifying the fibers means precisely giving oneself a bijection of this set. So coverings of the circle are classified by a set together with a bijection of this set, that is, by an action of the group $\mathbf Z$. –  ACL Nov 23 '11 at 13:32
    
With respect to the second sentence of the previous comment, note that the group of integers can be presented, in the category of groupoids, as the groupoid $$\mathbf I=\pi_1([0,1],\{0,1\}) $$ with $0,1$ identified. –  Ronnie Brown Dec 22 '11 at 14:42
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Re ACL's comment on coverings: One reason I got involved with groupoids was that in writing the first edition of my topology text in the 1960s I got annoyed at having to divert into covering theory to get this basic example of the fundamental group of the circle. After finding the use of groupoids, I also rewrote the basic theory of coverings using not actions, but that a covering {\bf map} of spaces is modelled by a covering {\bf morphism} of groupoids. I invite people to compare and contrast this exposition in "Topology and Groupoids" with those in other texts. –  Ronnie Brown Mar 31 '12 at 22:05

You mention Ronnie Brown, but have you looked up his book on topology and groupoids. I think what you ask for is there. The other very early source for this sort if calculation is in P. J. Higgins little monograph which is a TAC reprint at (http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html). There is a lot of stuff in there which you do not find most other places.

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I think the relevant formula is 8.4.1 in T&G. This is applied in section 9.2 to the Phragmen-Brouwer property and the Jordan Curve Theorem.

My original motivation for the investigation was to avoid a detour to compute the fundamental group of the circle: a basic theorem should compute THE basic example! I like the view of the integers (an infinite set) as an identification of a groupoid $\mathbf I $ with 4 arrows, identifying 0 and 1.

Also I tend to see covering spaces in terms of covering morphisms of groupoids, since then a covering map is algebraically modelled by a covering morphism, whereas an action is one step further.

In the new book `Nonabelian algebraic topology', published by the EMS, the van Kampen style arguments are used to compute relative homotopy groups as modules, and second relative homotopy groups as crossed modules, using colimit calculations.

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