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In algebraic topology, we show that a group $G$ acting properly discontinously on a simply connected (and sufficently connected) topological space $Y$ is isomorphic to the fundamental group of $Y/G$.

I found generalisations when $Y$ is a tree (if $G$ acts without inversion on a tree $Y$, then $G$ is isomorphic to the fundamental group of the graph of groups $Y/G$) and when $Y$ is an orbifold and $G$ is finite (if a finite group $G$ acts on a simply connected orbifold $Y$ then $G$ is isomorphic to the fundamental group of the orbifold quotient $Y/G$).

Do you know other generalizations?

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There is the theory of complexes of groups (which is a higher dimensional analog of graph of groups). See for example Bridson-Haefliger pp.546 –  HenrikRueping Aug 20 '12 at 10:11

2 Answers 2

up vote 2 down vote accepted

The main theorem in Chapter 11 of "Topology and Groupoids" is that if the group $G$ acts properly and discontinuously on a Hausdorff space $X$ which admits a universal cover, then the fundamental groupoid of the orbit space $X/G$ is isomorphic to the orbit groupoid $\pi_1(X)/\!/G$. (Any improvement suggested? The key point is that if a group acts on a space $X$ then it also acts on the fundamental groupoid $\pi_1 (X)$, so that the orbit groupoid is defined by an obvious universal property.) The proof goes by verifying the appropriate universal property. The problem of calculating this orbit groupoid is also dealt with there. As an example, the fundamental group of the symmetric square of a space is calculated. See also arXiv:math/0212271 for an account of some of this Chapter.

A basic paper on orbit groupoids is

J. Taylor, "Quotients of groupoids by the action of a group", Math. Proc. Camb. Phil. Soc., 103, (1988) 239-249.

Later: I should also mention the earlier work, but not using the language of groupoids, of M.A. Armstrong, in two papers:

"Lifting homotopies through fixed points", Proc. Roy. Soc. Edinburgh A93 (1982) 123--128. Also II 96 (1984) 201--205.

Ross Geoghegan in his review in MathSciNet of the second paper in 1986 writes: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years."

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Thank you for your answer. Do you know wether there exists a generalization for non properly discontinous actions but on specific spaces (like manifolds)? –  Seirios Aug 22 '12 at 15:49

@Seirios: To answer your comment, the precise conditions used in the book are:

(a) The projection $p : X \to X/G$ has the path lifting property: i.e. if $\bar{a} : I \to X/G$ is a path, then there is a path $a : I \to X$ such that $pa = \bar{a}$.

(b) If $x \in X$, then $x$ has an open neighbourhood $U_{x}$ such that

(i) if $g \in G$ does not belong to the stabiliser $G_{x}$ of $x$, then $U_{x} \cap (g\cdot U_{x}) = \varnothing$;

(ii) if $a$ and $b$ are paths in $U_{x}$ beginning at $x$ and such that $pa$ and $pb$ are homotopic rel end points in $X/G$, then there is an element $g \in G_{x}$ such that $g\cdot a$ and $b$ are homotopic in $X$ rel end points.

It would be interesting to find nice examples where these apply but not in the properly discontinuous situation! Condition (b)(ii) holds if $X$ is semilocally simply-connected.

You could also look at the papers of M.A. Armstrong in this area to see if they give clues as to possible generalisations from the properly discontinuous case. He eschews the use of groupoid notions, which to my mind makes the results seem more technical.

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