Let p: (X^,x^)--->(X,x) be a (based) covering space, both path connected and locally simply connected.

Define R[p] to be the set of  closed paths in  X at x which when lifted to x^ through p are closed in X^.

a)  show R[p] is closed under composition, reversing direction and contains the trivial path.

b) show R[p] is closed under homotopy of closed paths at x^ (that is, if f is in R[p] and g is homopic to f then g is in R[p])
Conclude the homotopy classes in R[p] define a subgroup  G of the fundamental group of X based at x.

c) show p induces an isomorphism between the fundamental group of X^ based at x^ with this subgroup.

d) for each point q^ in X^ lying over q in X choose a  fixed path in X^ between x^ and q^.
Show any  other path between x^ and q^ is obtained up to homotopy by first going around a closed path lifted from a representative of an element in G then doing the fixed path.

e) define two paths from x to q in X to be equivalent if they both lift to paths ending at the same point in X^.
 show the correspondence equivalence class goes to endpoint is a bijection between classes and points of X^.

f) use d) to deduce (X^,x^) is, via the bijection of e), homeomorphic as based covering spaces to the construction of the class last week of the covering X/G constructed from the subgroup G.


Recall the construction made in class:

Take a space X which is LSC, namely it has a basis B of simply connected open sets.  Let x belong to X. Let G be a subgroup of the fundamental group of X based at x. Recall a path connected space is simply connected if any two paths with the same endpoints are homotopic keeping the endpoints fixed.

Define an equivalence relation on paths in X starting at x by: two paths are equivalent iff  
1) they have the same endpoints in X and
2) the closed curve starting at x obtained by going out along one and then back along the other lies in a class belonging to G.

 We will define  a topology on the set of equivalence classes X^ so that the map p= (equivalence class goes to endpoint): X^ ---> X is a covering map. To do this partition the preimage by p of a basic open set U (that is, U is in the basis) in X into equivalence classes where two paths ending in U are equivalent if when connecting their endpoints by a path in U the resulting closed path belongs to a class in G.
It follows  directly from the definitions made (with real thinking though) that these equivalence classes as U varies form a basis of a topology on X^  and that with this topology X^ ---> X is a covering map.