Covering Space Actions

Subjects: Algebraic Topology
Links: Group Actions, Covering Maps, Automorphism Group of a Covering Map, Topological Groups, Topological Connectedness, Local Connectedness

Def: Suppose we are given an action by a group $\mathrm{\Gamma }$ on a topological space $E$. It is called a covering space action if $\mathrm{\Gamma }$ acts by homeomorphism and every point $e\in E$ has a neighbourhood $U$ satisfying the following condition: $$U \cap (g \cdot U) \neq \varnothing \iff g = 1.$$
We get an even stronger property, that all of its images under elements of $\mathrm{\Gamma }$ are pairwise disjoint: if $g,h\in \mathrm{\Gamma }$ are distinct elements, then $(g\cdot U)\cap (h\cdot U)=g\cdot (U\cap g{h}^{-1}\cdot U)=\varnothing$.

Let us note that if $\mathrm{\Gamma }$ acts on a topological space $E$ by homeomorphism, then there exists a group homomorphism $\phi :\mathrm{\Gamma }\to \text{Homeo}(E)$.

Obs: For any covering $q:E\to X$, the action ${\text{Aut}}_q(E)$ on $E$ is a covering space action.

Obs: Given a covering space action of a group $\mathrm{\Gamma }$ on a topological space $E$, then the restriction of the action to any subgroup of $\mathrm{\Gamma }$ is a covering space action.

Covering space actions are often called properly discontinuous actions.

Def: Given an action of a group $\mathrm{\Gamma }$ on a space $E$ by homeomorphism, each $g\in \mathrm{\Gamma }$ determines a homeomorphism from $E$ to itself by $e\mapsto g\cdot e$. We say that the action is effective if the identity of $\mathrm{\Gamma }$ is the only element for which this homeomorphism is the identity.

We see that every free action is effective. If $\mathrm{\Gamma }$ acts effectively, it is frequently useful to identify $\mathrm{\Gamma }$ with the corresponding group of homeomorphisms of $E$.

Using the group homomorphism because $\mathrm{\Gamma }$ acts on $E$ by homeomorphism, then the action of $\mathrm{\Gamma }$ is effective iff $\mathrm{\Phi }:\mathrm{\Gamma }\to \text{Homeo}(E)$ is injective.

Covering Space Quotient Theorem: Let $E$ be a connected, locally path-connected space, and suppose we are given an effective action of a group $\mathrm{\Gamma }$ on $E$ by homeomorphism. Then the quotient map $q:E\to E/\mathrm{\Gamma }$ is a covering map iff the action is a covering space action. In the case, $q$ is normal covering map, and ${\text{Aut}}_q(E)=\mathrm{\Gamma }$, considered as a group of homeomorphisms of $E$.

Prop: Let $\mathrm{\Gamma }$ be a discrete subgroup of a connected and locally path-connected topological group $G$. Then the action of $\mathrm{\Gamma }$ on $G$ by left translations is a covering map space action, so the quotient map $q:G\to G/\mathrm{\Gamma }$ is a normal covering map.

Cor: Suppose $G$ and $H$ are connected and locally path-connected topological groups, and $\phi :G\to H$ is a surjective continuous homomorphism with discrete kernel. If $\phi$ is an open or closed map, then it is a normal covering map.

Prop: Let $E$ be a Hausdorff space. Every free, continuous action of a finite group on $E$ is a covering space action with Hausdorff quotient.