By Frederick S. Woods

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18, (θn ) is an inﬁnite cyclic group. 25. For m = n, the groups Bm and Bn are not isomorphic. Proof. 24 implies that the image of Z(Bn ) in Bn /[Bn , Bn ] ∼ = Z is a subgroup of Z of index n(n − 1). If Bm is isomorphic to Bn , then we must have m(m − 1) = n(n − 1), and hence m = n. 1. 7). 2. Verify that Δ2n = (σ1 σ2 · · · σn−1 )n . 3. Verify that Pn is the minimal normal subgroup of Bn containing σ12 = A1,2 . 4. 4) using only the expression of Ai,j via σ1 , . . , σn−1 and the braid relations between these generators.

Integral powers of diﬀeomorphisms and their products are also diﬀeomorphisms. Therefore the surjectivity of η implies the following assertion. 34. An arbitrary self-homeomorphism of the pair (D, Qn ) is isotopic in the class of self-homeomorphisms of this pair to a diﬀeomorphism (D, Qn ) → (D, Qn ). 1. 2. (a) Consider an embedded r-gon P ⊂ M (with r ≥ 3) meeting Q precisely in its vertices. Moving along ∂P in the direction provided by the orientation of M , we meet consecutively r edges, say α1 , α2 , .

U0n to u1 , . . , un , respectively. Observe that c−1 (u0 ) is the closed subgroup of Top(M ) consisting of all f ∈ Top(M ) such that f (u0i ) = u0i for i = 1, 2, . . , n. The formula (u, f ) → θu f deﬁnes a homeomorphism U × c−1 (u0 ) → c−1 (U ) commuting with the projections to U . The inverse homeomorphism sends any g ∈ c−1 (U ) to the pair (c(g), (θc(g) )−1 g) ∈ U × c−1 (u0 ). 36. Two elements of Top(M ) have the same image under the evaluation map e if and only if they lie in the same left coset of Top(M, Q) in Top(M ).