# An Introduction to Lorentz Surfaces (De Gruyter Expositions by Tilla Weinstein By Tilla Weinstein

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The product of this path with g is a lift of f g : I → Cn ending at idD and beginning at f (0) g(0). Thus, ∂(βγ) = [f (0) g(0)] = [f (0)] [g(0)] = ∂(β) ∂(γ) . 4. The homomorphism ∂ : Bn = π1 (Cn , q) → M(D, Q) can be described in terms of parametrizing isotopies as follows. 2, ∂(β) ∈ M(D, Q) is the isotopy class of f0 : (D, Q) → (D, Q). 3. It suﬃces to verify that ∂ and η coincide on the generators σi , where i = 1, 2, . . , n − 1. Since η(σi ) = ταi , we need only check that ∂(σi ) = ταi . Let {gt : D → D}t∈I be the isotopy of the identity map g0 = id : D → D into g1 = ταi obtained by rotating αi in D about its midpoint counterclockwise.

4. The homomorphism ∂ : Bn = π1 (Cn , q) → M(D, Q) can be described in terms of parametrizing isotopies as follows. 2, ∂(β) ∈ M(D, Q) is the isotopy class of f0 : (D, Q) → (D, Q). 3. It suﬃces to verify that ∂ and η coincide on the generators σi , where i = 1, 2, . . , n − 1. Since η(σi ) = ταi , we need only check that ∂(σi ) = ταi . Let {gt : D → D}t∈I be the isotopy of the identity map g0 = id : D → D into g1 = ταi obtained by rotating αi in D about its midpoint counterclockwise. Then {ft = g1−t : D → D}t∈I is an isotopy of f0 = τα into f1 = id.

By a spanning arc on (M, Q), we mean a subset of M homeomorphic to I = [0, 1] and disjoint from Q ∪ ∂M except at its two endpoints, which should lie in Q. , have no self-intersections. Let α ⊂ M be a spanning arc on (M, Q). The half-twist τα : (M, Q) → (M, Q) is obtained as the result of the isotopy of the identity map id : M → M rotating α in M about its midpoint by the angle π in the direction provided by the orientation of M . The half-twist τα is the identity outside a small neighborhood of α in M .

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