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Consider the following situation. Let p : T −→ X be a covering space, x0 ∈ X , x0 ∈ p−1 (x0 ) ∈ T . Let f : Z −→ X be a map, so that f (z0 ) = x0 . There is a natural question: Question: Does there exist a map f : Z −→ T covering the map f : Z −→ X , such that f (z0 ) = x0 ? In other words, the lifting map f should make the following diagram commutative: T f (20) Z f ✒ p ❄ ✲ X NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 55 where f (z0 ) = x0 , f (z0 ) = x0 . 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 )).

We already have a barycentric subdivision of each j -the side ∆qj by (q − 1)-simplices (1) (n) ∆j , . . , ∆j , n = q!. The cones over these simplices, j = 0, . . , q , with a vertex x∗ constitute a barycentric subdivision of ∆q . 8. Let V ⊂ U be two open sets of Rn such that their closure V¯ , U¯ are compact sets and V¯ ⊂ U . Then there exists a finite triangulation of V by n-simplices {∆n (i)} such that ∆n (i) ⊂ U . Proof. For each point x ∈ V¯ there exists a simplex ∆n (x) with a center at x and ∆n (x) ⊂ U .

11. What does it mean geometrically that a pair (X, A) is 0-connected? 1connected? Give some alternative description. 12. Let (X, A) be an n-connected pair of CW -complexes. Prove that (X, A) is homotopy equivalent to a CW -pair (Y, B) so that B ⊂ Y (n) . NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 43 6. 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.

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