By Botvinnik B.

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The group π1 (Mg2 (1)) is isomorphic to a group on generators c1 , . . , c2g+1 wit a single relation c21 · · · c22g+1 = 1. 2. The group π1 (Mg2 (2)) is isomorphic to a group on generators c1 , . . , c2g+2 wit a single relation c21 · · · c22g+1 c22g+2 = 1. 13. 7. 14. Compute π1 (RPn ), π1 (Kl2 ). 15. Compute the group π/[π, π] for the groups π = π1 (Mg2 (1)), π1 (Mg2 (2)). 16. 7 are pairwise nonisomorphic. Prove that any two manifolds above are not homeomorphic and even are not homotopy equivalent to each other.

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 )).

Let a base point x0 ∈ Y1 ∩ Y2 ⊂ X , and ρ1 : π1 (Y1 ) −→ π1 (X), ρ2 : π1 (Y2 ) −→ π1 (X). Then π1 (X) ∼ = π1 (Y1 ) ∗π (Z) π1 (Y2 ). 20. 8 in the case of finite CW -complexes using induction on the number of cells of Y1 ∩ Y2 . Remark. There is more general version of Van Kampen Theorem, see [Massey, Ch. 2]. 52 BORIS BOTVINNIK 7. 1. Definition and examples. A path-connected space T is a covering space over a pathconnected space X , if there is a map p : T −→ X such that for any point x ∈ X there exists a path-connected neighbourhood U ⊂ X , such that p−1 (U ) is homeomorphic to U × Γ (where Γ is a discrete set), futhermore the following diagram commutes ∼ = p−1 (U ) ❅ (18) ❅ ✲ U ×Γ p ❅ ❅ ❘ ❅ ✠ pr U The neighbourhood U from the above definition is called elementary neighborhood.