Overview
By defining a (well-behaved) action of the fundamental group on a cover we can use many of the tools of group theory to understand the topology of the spaces.
The main results of this idea are that we can learn about automorphisms that preserve the base space from group action automorphisms, or vice versa.
Definition
Let
where
This defines a group action of
Proof
First, as a sanity check we can make sure that the action makes sense
Next, we check the axioms for a group action.
-
Identity action
-
Associative because the path lifts uniquely. Thus,
Transitive action
Proof
Equivariance of cover automorphisms
Fix
Proof
Correspondence between automorphisms
Proof
todo We can look at the map
Just need to show that it is injective and surjective.
Corollaries
- If
is a regular cover,
- If
is a universal cover,