Covering Fundamental groups

Lemma

Let be a covering map with , .
Then is injective
(where is the natural homomorphism induced by ).

Definition

The (right) group action of fundamental group on
is defined as:

where is the Path Push Forward of .

Lemma

Let be a covering map, path connected.
Then:

1. acts transitively on iff is path connected
2. Stabiliser of is
3. If is path connected, have bijection:

induced by on

Corollary

Suppose is a universal cover.
Then each point determines a bijection
given by .

Proof

Immediate from previous lemma.