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Theorem of Seifert and Van Kampen. 2] for detailes. Let G1 , G2 be two groups with system of generators A1 , A2 and relations R1 , R2 respectively. A group with a system of generators A1 ∪ A2 (disjoint union) and system of relations R1 ∪ R2 is called a free product of G1 and G2 and is denoted as G1 ∗ G2 . 17. Prove that the group Z2 ∗ Z2 contains a subgroup isomorphic to Z and (Z2 ∗ Z2 )/Z ∼ = Z2 . 18. Let X , Y be two CW -complexes. Prove that π1 (X ∨ Y ) = π1 (X) ∗ π1 (Y ), where the base points x0 ∈ X and y0 ∈ Y are identified with a base point in X ∨ Y .

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. 0 1 00000 000 00000 000 11111 111 11111 111 0 1 00000 11111 000 111 00000 11111 000 00000 000111 00000 000 11111 111 11111 111 00000 000111 00000 000 11111 111 00000 11111 00011111 111 00000 11111 000 111 00000 11111 000111 111 00000 11111 000 00000 11111 000 111 00000 11111 000 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 11111 000 111 00000 11111 000 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 00000 11111 000111 111 00000 11111 000 111 00000 11111 000 111 00000 11111 000 111 00000 11111 000 111 00000 11111 000 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 00000 000111 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 111 00000 000 00000 000 11111 111 11111 00000 000111 00000 000 11111 111 11111 111 The map η is an identical map on (Dn+1 × {0}) ∪ (S n × I).

6. 5. Two coverings T1 −→ X and T2 −→ X are isomorphic if and only if for any (1) (2) (1) (2) −1 two points x0 ∈ p−1 1 (x0 ), x0 ∈ p2 (x0 ) the subgroups (p1 )∗ (π1 (T1 , x0 )) (p1 )∗ (π1 (T2 , x0 )) belong to the same conjugation class. 8. 5. Let H ⊂ G be a subgroup. Recall that a normalizer N (H) of H is a maximal subgroup of G such that H is a normal subgroup of that group. The subgroup N (H) of the group G may be described as follows: N (H) = g ∈ G | gHg −1 = H . NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 57 Recall also that the group π1 (X, x0 ) acts on the set Γ = p−1 (x0 ), and Γ may be considered as a right π1 (X, x0 )-set; the subgroup p∗ (π1 (T, x0 )) is the “isotropy group” of the point x0 ∈ p−1 (x0 ).