By Botvinnik B.

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Then the path f (ω) = γ has a unique lift γ so that γ(0) = x0 . We define f (z) = γ(1) ∈ T . We have to check that the construction does not depend on the choice of ω . Let ω ′ be another path such that ω ′ (0) = z0 , ω ′ (1) = z , see Fig. 6. ω′ z z0 f (z) γ′ ω x0 = f (z0 ) γ Figure 24 Let γ ′ = f (ω ′ ). Then we have a loop β = (γ ′ )−1 γ , and [β] ∈ f∗ (π1 (Z, z0 )). Since f∗ (π1 (Z, z0 )) ⊂ p∗ (π1 (T, x0 )), the loop β may be lifted to the loop β in T . In particular, it follows that γ(1) = γ ′ (1) because of uniqueness of the liftings γ and γ ′ and γ .

Let αi be the generators of π1 (X (1) , x0 ) corresponding to the the cells e1i , and βj ∈ π1 (X (1) , x0 ) = F (αi | i ∈ I) be elements determined by the attaching maps fj : S 1 −→ X 1 of the cells e2j . Then 1. π1 (X, x0 ) ∼ = π1 (X (2) , x0 ); 2. π1 (X, x0 ) is a group on generators αi , i ∈ I , and relations βj = 1, j ∈ J . Proof. We consider the circle S 1 as 1-dimensional CW -complex. e. the homomorphism ι∗ : π1 (X (1) , x0 ) −→ π1 (X, x0 ) induced by the inclusion ι : X (1) −→ X , is an epimorphism.

Then any map Y −→ X is homotopic to a constant map. The same statement holds for “pointed” spaces and “pointed” maps. 9. 9 using the Cellular Approximation Theorem. Remark. For each pointed space (X, x0 ) define πk (X, x0 ) = [S k , X] (where we consider homotopy classes of maps f : (S k , s0 ) −→ (X, x0 )). Very soon we will learn a lot about πk (X, x0 ), in particular, that there is a natural group structure on πk (X, x0 ) which are called homotopy groups of X . 10. The homotopy groups πk (S n ) are trivial for 1 ≤ k < n.