By Botvinnik B.

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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. 6. Theorem of Seifert and Van Kampen. 2] for detailes.

Here we need the following lemma. 6. (Free-point-Lemma) Let U be an open subset of Rp , and ϕ : U → D be a ◦q continuous map such that the set V = ϕ−1 (dq ) ⊂ U is compact for some closed disk dq ⊂ D . ◦q If q > p there exists a continuous map ψ : U −→ D such that 1. ψ|U \V = ϕ|U \V ; 2. e. there exists a point y0 ∈ dq \ ψ(U ). We postpone a proof of this Lemma for a while. Remark. 6 are homotopic relatively to U \ V : it is ◦q enough to make a linear homotopy: ht (x) = (1 − t)ϕ(x) + tψ(x) since the disk D is a convex set.

4. Any continuous map f : X −→ Y of CW -complexes is homotopic to a cellular map. 5. Let f : X −→ Y be a continuous map of CW -complexes, such that a restriction f |A is a cellular map on a CW -subcomplex A ⊂ X . Then there exists a cell map g : X −→ Y such that g|A = f |A and, moreover, f ∼ g rel A. First of all, we should explain the notation f ∼ g rel A which we are using. Assume that we are given two maps f, g : X −→ Y such that f |A = g|A . A notation f ∼ g rel A means that there exists a homotopy ht : X −→ Y such that ht (a) does not depend on t for any a ∈ A.