What means act via deck transformations?

Question

I am not sure what is does it menas for a group to act via deck transformations. I know the concepts of a fundamental group and a acts via homeomorphism, but I want an intuition (or a formal idea) of what is that the fundamental group acts via deck transformations. Thanks

Answer 1

An action of a group $G$ on a topological space $X$ is simply a homomorphism $\mathcal A : G \to \text{Homeo}(X)$ where $\text{Homeo}(X)$ is the group of homeomorphisms from $X$ to itself, with the operation of composition.

If you are given any covering map $f : X \mapsto Y$, the set $D_f$ of deck transformations is literally a subgroup $D_f < \text{Homeo}(X)$, and the action of this subgroup is literally the inclusion homomorphism $D_f \hookrightarrow \text{Homeo}(X)$. So far the "action" terminology doesn't seem helpful.

However, if $Y$ is path connected and $X$ is simply connected then there is an isomorphism $\gamma : \pi_1(Y,y) \approx D_f$; this is perhaps the fundamental theorem of covering space theory. In this situation the action terminology is more meaningful: the composition $$\mathcal A : \pi_1(Y,y) \xrightarrow{\gamma} D_f \hookrightarrow \text{Homeo}(X) $$ is an action, and we can refer it as the "action of $\pi_1(Y,y)$ on $X$ via deck transformations".