Algebraic topology notes by Botvinnik B.

By Botvinnik B.

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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. Let K1 be a union of all simplices ∆q such that the intersection ϕ(∆) ∩ d4 is not empty.

3. 4. Let T1 −→ X , T2 −→ X be two coverings, x0 ∈ X , x0 ∈ p1 (x0 ), x0 ∈ (1) (2) p2 (x0 ). There exists a morphism ϕ : T1 −→ T2 such that ϕ(x0 ) = x0 if and only if (1) (2) (p1 )∗ (π1 (T1 , x0 )) ⊂ (p1 )∗ (π1 (T2 , x0 )). 5. 4. A morphism ϕ : T −→ T is automorphism if there exists a morphism ψ : T −→ T so that p ψ ◦ ϕ = Id and ϕ ◦ ψ = Id. Now consider the group Aut(T −→ X) of automorphisms of a given covering p : T −→ X . The group operation is a composition and the identity element p is the identity map Id : T −→ T .

We have to define an extension of F1 from the side g(S n ) × I and the bottom base g(Dn+1 ) × {0} to the cylinder g(Dn+1 ) × I . By definition of CW -complex, it is the same as to construct an extension of the map ψ = F (n) ◦ g : (Dn+1 × {0}) ∪ (S n × I) −→ Y to a map of the cylinder ψ ′ : Dn+1 × I −→ Y . Let η : Dn+1 × I −→ (Dn+1 × {0}) ∪ (S n × I) be a projection map of the cylinder Dn+1 × I from a point s which is near and a bit above of the top side Dn+1 × {1} of the cylinder Dn+1 × I , see the Figure below.

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