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The subgroup p∗ (π1 (T, x0 )) ⊂ π1 (X, x0 ) is called the covering p group of T −→ X . Let x′0 = x0 , p(x′0 ) = p(x0 ) = x0 . Consider a path α : I −→ T such that α(0) = x0 , α(1) = x′0 . Then the projection α = p(α) is a loop in X , see Figure to the left. Clearly α# : p∗ (π1 (T, x0 )) −→ p∗ (π1 (T, x′0 )) given by α# (g) = αgα−1 is an isomorphism. Consider the coset π1 (X, x0 )/p∗ (π1 (T, x0 )) (the subgroup p∗ (π1 (T, x0 )) ⊂ π1 (X, x0 ) is not normal subgroup in general). 1. There is one-to-one correspondence p−1 (x0 ) ←→ π1 (X, x0 )/p∗ (π1 (T, x0 )).

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. The group π1 (Mg2 (1)) is isomorphic to a group on generators c1 , . . , c2g+1 wit a single relation c21 · · · c22g+1 = 1.

The properties (a), (b), (e) follow from the definition. 10. 3. Let ǫi ∈ H σi be a vector which has σi -coordinate equal to 1, and all others are zeros. Thus (ǫ1 , . . , ǫk ) ∈ E(σ). For each k -frame (v1 , . . , vk ) ∈ E(σ) consider the transformation: (13) T = Tǫk ,vk ◦ Tǫk−1 ,vk−1 ◦ · · · · · · ◦ Tǫ1 ,v1 : Rn −→ Rn σi First we notice that vi = −ǫi since vi ∈ H . Thus the transformations Tǫi ,vi are well-defined. 11. Prove that the transformation T takes the k -frame (ǫ1 , . . , ǫk ) to the frame (v1 , .

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