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

By Tilla Weinstein

Show description

Read Online or Download An Introduction to Lorentz Surfaces (De Gruyter Expositions in Mathematics 22) PDF

Similar mathematics books

The Mathematics of Paul Erdos II (Algorithms and Combinatorics 14)

This can be the main complete survey of the mathematical lifetime of the mythical Paul Erd? s, essentially the most flexible and prolific mathematicians of our time. For the 1st time, all of the major components of Erd? s' examine are lined in one venture. due to overwhelming reaction from the mathematical neighborhood, the undertaking now occupies over 900 pages, prepared into volumes.

Extra resources for An Introduction to Lorentz Surfaces (De Gruyter Expositions in Mathematics 22)

Example text

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 .

Download PDF sample

Rated 4.56 of 5 – based on 16 votes