 Free action of a group

A group action is free when only the identity element fixes any given point, equivalently when every stabilizer subgroup is trivial. The trivial group acts freely on every set. Each nonidentity element in a free group action induces a fixed-point-free bijection; the fixed-point property of even-dimensional complex projective space obstructs such actions by homeomorphisms.

For a finite group acting freely by homeomorphisms on a Hausdorff manifold, the quotient map is a covering map. Around a point choose a small neighborhood disjoint from all its nonidentity translates, using finiteness and the Hausdorff property. Its translates are the disjoint sheets above its image in the quotient. For an infinite group, freeness alone does not supply such neighborhoods.