Galois Correspondence

We have seen that is a path connected covering space and the map is injective

So a subgroup

If is another basepoint of , as path-connected, can choose a path . Then is a loop in based at , and ie the two subgroups obtained are conjugate.

So fixing we have functions

and

Want to show that imposing appropriate equivalence relations of LHSs, makes those into bijections.

Proposition

Suppose that is path-connected, Locally Path Connected, and Semi-locally Simply Connected. Then for any subgroup there is a covering space with

Proof

Let be the Universal Cover that we constructed, whose underlying set is the set of homotopy classes of paths in starting at . Define on by iff and This is an equivalence relation

1. as ( contains the identity element)
2. then so its inverse
3. then and so the product . So is an equivalence relation on

Define and to be the induced map

If and and . then and are identified by as for any path in . So is a covering map ()

Need If then the lift of to starts at and ends at .

So the lift to ends at , so it is a loop. So . So we have .

Conversely, if then the lift of to starts at and ends at .

As is a loop in we have ie . This shows .

Proposition (Based uniqueness)

Suppose that is path-connected, Locally Path Connected, and Semi-locally Simply Connected. If , are path connected covering spaces, there is a homomorphism st iff

Proof

If exists then and is an isomorphism, so

For the other direction, let . Will show

Consider with the end point of the lift to which starts at . Now iff and end at the same point iff iff so the induced map is a continuous bijection and an open map as both covering maps to are local homeomorphisms.

Proposition (Unbased uniqueness)

If and are path connected covering spaces, then there is a homeomorphism st iff is conjugate to in