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A necessary condition is clearly that f * (p 1 ( Z , z 0 )) Õ p * (p 1 ( Y , y 0 )). The amazing fact is
that this condition is also sufficient provided that some weak connectivity conditions
hold.
Definition. A topological space X is said to be locally path-connected if every neigh-
borhood of a point contains a neighborhood that is path-connected.
Fortunately, the spaces of interest to us are locally path-connected. Manifolds are
trivially locally path-connected, but so are CW complexes (use induction on the
number of cells).
7.4.2.11. Theorem. Let ( Y ,p) be a covering space for a space X . Let Z be a path-
connected and locally path-connected space. Let x 0 ΠX , y 0 Πp -1 ( x 0 ), and z 0 ΠZ . Then
a map f : ( Z , z 0 ) Æ ( X , x 0 ) lifts to a map ˜
: ( Z , z 0 ) Æ ( Y , y 0 ) if and only if f * (p 1 ( Z , z 0 )) Õ
p * (p 1 ( Y , y 0 )).
Proof. See [Mass67] or [Jäni84]. The diagram below should help clarify what is
being said:
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We can deduce a number of important results from Theorem 7.4.2.11.
7.4.2.12. Theorem. Let ( Y 1 ,p 1 ) and ( Y 2 ,p 2 ) be covering spaces for a space X , where
Y 1 and Y 2 are path-connected and locally path-connected spaces. Let x 0 Œ X and y i Œ
p i -1 ( x 0 ). The two covering spaces are isomorphic via a bundle isomorphism f : ( Y 1 , y 1 )
Æ ( Y 2 , y 2 ) if and only if p 1* (p 1 ( Y 1 , y 1 )) = p 2* (p 1 ( Y 2 , y 2 )).
Proof.
See [Mass67]. The following diagram might again help:
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To get the next theorem we need another technical definition.
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