By Botvinnik B.

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**Example text**

1. General definitions. Here we define the homotopy groups πn (X) for all n ≥ 1 and examine their basic properties. Let (X, x0 ) be a pointed space, and (S n , s0 ) be a pointed sphere. We have defined the set [S n , X] as a set of homotopy classes of maps f : S n −→ X , such that f (s0 ) = x0 , and homotopy between maps should preserve this property. In different terms we can think of a representative of [S n , X] as a map I n −→ X such that the image of the boundary ∂I n of the cube I n maps to the point x0 .

By compactness of V¯ there exist a finite number of simplices ∆n (xi ) covering V¯ . 6 to conclude that a union of finite number of ∆n (xi ) has a finite triangulation. 5. 6. We consider carefully our map ϕ : U −→ D . First we construct the disks d1 , d2 , d3 , d4 inside the disk d with the same center and of radii r/5, 2r/5, 3r/5, 4r/5 respectively, where r is a radius of d. Then we cover V = ϕ−1 (d) by finite number of p-simplexes ∆p (j), such that ∆n (j) ⊂ U . Making, if necessary, a barycentric subdivision (a finite number of times) of these simplices, we can assume that each simplex ∆p has a diameter d(ϕ(∆p )) < r/5.

Examples. 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 .