Proposition 4.1 (from Hatcher): A covering space projection induces isomorphisms for all .
We will elaborate more on this proposition in this blog post. Basically, we will need to show that is a homomorphism and also bijective (surjective and injective).
, which we can see is the same.
Thus, is a homomorphism.
For surjectivity, we need to use a certain Proposition 1.33: Suppose given a covering space and a map with path-connected and locally path-connected. Then a lift of exists iff .
Let , where . Since is simply connected for , . Thus . By Proposition 1.33, a lift of exists, where .
i.e. we have . Hence is surjective.
Let , where with a homotopy of to the trivial loop .
By the covering homotopy property (homotopy lifting property), there exists a unique homotopy of that lifts , i.e. . There is a lifted homotopy of loops starting with and ending with a constant loop. Hence in and thus is injective.