nLab
orbit groupoid

If a group GG acts on a groupoid Γ\Gamma, then there is an orbit groupoid Γ//G\Gamma /\! / G which is a groupoid with trivial GG-action and with a GG-morphism ΓΓ//G\Gamma \to \Gamma /\! / G universal for GG-morphisms to groupoids with trivial GG-action. So in principle Γ//G\Gamma /\! / G is obtained from Γ\Gamma by identifying g.γg.\gamma and γ\gamma for all gGg \in G, γΓ\gamma \in \Gamma.

Note that if a group GG acts on a space XX then it has an induced action on the fundamental groupoid Π 1X\Pi_1 X, and so there is an induced morphism

α:Π 1X//GΠ 1(X/G).\alpha: \Pi_1 X /\! / G \to \Pi_1 (X/G).

So there is interest in when this morphism is an isomorphism.

α\alpha is an isomorphism if XX is Hausdorff, has a universal cover, and the action of GG on XX is discontinuous.

This allows some calculation of the fundamental groups of orbit spaces.

References

  • R. Brown, Topology and groupoids, Booksurge, 2006, Chapter 11.

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

Revised on April 8, 2009 00:17:16 by Toby Bartels (71.104.234.95)