By Botvinnik B.

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E. p ◦ Ft (A) = x0 . It follows that p ◦ Ft = ht ◦ p, where ht : X/A −→ X/A is some homotopy, such that h0 = IdX/A and h1 = p ◦ q ; it means that p ◦ q ∼ IdX/A . 3. Let X be a CW -complex and A ⊂ X be its subcomplex. Then X/A is homotopy equivalent to the complex X ∪ C(A), where C(A) is a cone over A. 2. 3. 2. Cellular Approximation Theorem. Let X and Y be CW -complexes. Recall that a map f : X −→ Y is a cellular map if f (X (n) ) ⊂ Y (n) for every n = 0, 1, . .. We emphasize that it is not required that the image of n-cell belongs to a union of n-cells.

11. 6. 12. For a group π , we let [π, π] be its commutator. Compute the group π/[π, π] for π = π1 (Mg ). Remark. We note that in particular π1 (T 2 ) ∼ = Z ⊕ Z, which is obvious from the product formula π1 (X × Y ) ∼ = π1 (X) × π1 (Y ). Recall that a non-oriented two-dimensional manifold of genus g is heomeomorphic either to Mg2 (1), a connective sum of a projective plane RP2 and g tori T 2 # · · · #T 2 , or to Mg2 (2), a connective sum of the Klein bottle Kl2 and g tori T 2 # · · · #T 2 . 7. 1.

10. Prove that a covering space p : T −→ X is regular if and only if there is no loop in X which is covered by a loop and a path (starting and ending in different points) in the same time. 11. Prove that if a covering space p : T −→ X is regular then there exists a free action of the group G = π1 (X, x0 )/π1 (T, x0 ) on the space T such that X ∼ = T /G. 12. Prove that a two-folded covering space p : T −→ X is always a regular one. 7. Let X be a “good” path-connected space (in particular, CW -complexes are “good” spaces), x0 ∈ X .