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.