By Boris Botvinnik

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1. p : R −→ S 1 , where S 1 = {z ∈ C | |z| = 1 }, and p(ϕ) = eiϕ . 2. p : S 1 −→ S 1 , where p(z) = z k , k ∈ Z, and S 1 = {z ∈ C | |z| = 1 }. 3. p : S n −→ RPn , where p maps a point x ∈ S n to the line in Rn+1 going through the origin and x. 2. Theorem on covering homotopy. The following result is a key fact allowing to classify coverings. 1. Let p : T → X be a covering space and Z be a CW -complex, and f : Z → X , f : Z → T such that the diagram T f (19) p ❄ ✲ X f Z ✒ commutes; futhermore it is given a homotopy F : Z × I −→ X such that F |Z×{0} = f .

Clearly the diagram (20) gives the following commutative diagram of groups: π1 (T, x0 ) f∗ (21) ✒ p∗ ❄ f∗ ✲ π1 (Z, z0 ) π1 (X, x0 ) It is clear that commutativity of the diagram (21) implies that f∗ (π1 (Z, z0 )) ⊂ p∗ (π1 (T, x0 )). (22) Thus (22) is a necessary condition for the existence of the map f . It turns out that (22) is also a sufficient condition. 4. Let p : T −→ X be a covering space, and Z be a path-connected space, x0 ∈ X , x0 ∈ T , p(x0 ) = x0 . Given a map f : (Z, z0 ) −→ (X, x0 ) there exists a lifting f : (Z, z0 ) −→ (T, x0 ) if and only if f∗ (π1 (Z, z0 )) ⊂ p∗ (π1 (T, x0 )).

Then the projection α = p(α) is a loop in X , see Figure to the left. Clearly α# : p∗ (π1 (T, x0 )) −→ p∗ (π1 (T, x′0 )) given by α# (g) = αgα−1 is an isomorphism. Consider the coset π1 (X, x0 )/p∗ (π1 (T, x0 )) (the subgroup p∗ (π1 (T, x0 )) ⊂ π1 (X, x0 ) is not normal subgroup in general). 1. There is one-to-one correspondence p−1 (x0 ) ←→ π1 (X, x0 )/p∗ (π1 (T, x0 )). Proof. Let [γ] ∈ π1 (X, x0 ), where γ : I −→ X , γ(0) = γ(1) = x0 . There exists a unique lifting γ : I −→ T of γ , so that γ(0) = x0 .