Covering map

G-coverings

Let G be a discrete groupacting on the topological spaceX. This means that each element g of G is associated to a homeomorphism H_g of X onto itself, in such a way that H_gh is always equal to H_g${\displaystyle \circ }$ H_h for any two elements g and h of G. (Or in other words, a group action of the group G on the space X is just a group homomorphism of the group G into the group Homeo(X) of self-homeomorphisms of X.) It is natural to ask under what conditions the projection from X to the orbit spaceX/G is a covering map. This is not always true since the action may have fixed points. An example for this is the cyclic group of order 2 acting on a product X × X by the twist action where the non-identity element acts by (x, y) ↦ (y, x). Thus the study of the relation between the fundamental groups of X and X/G is not so straightforward.