By Adem A., Milgram R.J., Ravenel D.C. (eds.)
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Extra resources for Homotopy Theory and Its Applications
Example text
3, [74] pp. 83–85). Consider now a hexagon G ⊂ H with sides a1 , b1 , a2 , b2 , a3 , b3 parameterized on the unit interval [0, 1]. Let now G′ := {x + iy ∈ C | x − iy ∈ G} be a copy of G in the negative half-plane H − . Denote the corresponding sides by a1′ , b1′ , a2′ , b2′ , a3′ , b3′ . We equip H − with the complex structure −i and the metric y −2 geucl . We construct a surface Y by identifying the points ak (t) with ak′ (t) for k = 1, 2, 3 and 0 ≤ t ≤ 1 (see Fig. 9). The complex structures i on G and −i on G′ fit together.
There is exactly one such geodesic for each x. The function x → dist α3 (x), α2 , x ≥ sup Re(α2 ) is strictly monotone increasing (show this as an exercise). We choose x in such a way that dist(α3 (x), α2 ) = ℓ2 . Then x and α3 (x) are uniquely determined by ℓ2 and α2 . 31). If ℓ2 = 0 then we choose x = sup Re(α2 ) and b2 = {x}. Let now b3 be the shortest geodesic segment connecting the point α3 ∩ Γ with the imaginary axis. By construction of Γ , we already have ℓ(b3 ) = ℓ3 . In the case where ℓ3 = 0 the ray Γ coincides with iR+ .
52 Let Y be a pair of pants with boundary components γ1 , γ2 , γ3 and lengths ℓ(γ1 ), ℓ(γ2 ), ℓ(γ3 ) ≥ 0. Then ℓ(γ1 ), ℓ(γ2 ), ℓ(γ3 ) determine Y up to isometry. e. the lengths ℓ1 , ℓ2 , ℓ3 of the sides b1 , b2 , b3 determine G uniquely up to orientation preserving isometry. Indeed, any pair of pants can be obtained by gluing two congruent hexagons together along three of their geodesic boundary segments. We will give an explicit geometric construction of a hexagon with prescribed lengths for b1 , b2 , b3 , and this construction leads to a unique hexagon up to isometry (see Fig.