The peripheral element at infinity is a loop around infinity #
The peripheral element periphInf of π₁(ℂ ∖ {0, 1}, 1/2) is defined as
(periph1 * periph0)⁻¹, so that the three peripheral elements have product one. This file proves
that it is what its name says: the class of a loop around the third puncture ∞.
Let δ be the circle |z| = 3, traversed counterclockwise once from the point
p₊ = 1/2 + (√35/2)·i, and let α₊ be the vertical segment from the basepoint 1/2 up to p₊.
The circle δ separates the punctures 0 and 1 from ∞, and the main theorem is
as paths in ℂ ∖ {0, 1}, where γ0 and γ1 are the peripheral loops around 0 and 1.
Consequently periph1 * periph0 is the class of α₊ · δ · α₊.symm, and periphInf is the class of
the circle |z| = 3 traversed clockwise in the affine coordinate z, transported to the
basepoint along α₊. In the chart w = 1/z at ∞ the same circle runs counterclockwise.
The anharmonic self-homeomorphism z ↦ z / (z − 1) of ℂ ∖ {0, 1} fixes the puncture 0 and
exchanges 1 with ∞, so it gives a second description of periphInf: the image of the loop γ1
around 1. The map moves the basepoint 1/2 to −1; transporting back along the path α₋₁ from
−1 to 1/2 through the closed upper half-plane, it induces an automorphism mob1InfMulAut of
π₁(ℂ ∖ {0, 1}, 1/2), and
mob1InfMulAut periph0 = periph0, mob1InfMulAut periph1 = periphInf.
These values are what identify pulling covers back along z ↦ z / (z − 1) with the exchange of
the branch points 1 and ∞ on permutation triples.
Main declarations #
TauCeti.ThricePuncturedSphere.pPlus: the point1/2 + (√35/2)·iof the circle|z| = 3.TauCeti.ThricePuncturedSphere.αPlus: the vertical segment from1/2topPlus.TauCeti.ThricePuncturedSphere.δ: the circle|z| = 3, counterclockwise frompPlus, withnorm_coe_δ.TauCeti.ThricePuncturedSphere.αPlus_trans_δ_trans_symm_homotopic_γ0_trans_γ1:α₊ · δ · α₊.symm ≃ γ0 · γ1.TauCeti.ThricePuncturedSphere.periph1_mul_periph0_eq_fromPath,TauCeti.ThricePuncturedSphere.periphInf_eq_fromPath:periph1 * periph0is the class ofα₊ · δ · α₊.symm, andperiphInfis the class ofα₊ · δ.symm · α₊.symm.TauCeti.ThricePuncturedSphere.αMob1Inf: the path from−1to1/2through the upper half-plane, withrange_αMob1Inf.TauCeti.ThricePuncturedSphere.mob1InfMulAut: the automorphism ofπ₁(ℂ ∖ {0, 1}, 1/2)induced byz ↦ z / (z − 1)andα₋₁, withmob1InfMulAut_periph0,mob1InfMulAut_periph1andmob1InfMulAut_periphInf, and the valuesmob1InfMulAut_symm_periph0andmob1InfMulAut_symm_periph1of its inverse.
References #
- E. Girondo and G. González-Diez, Introduction to Compact Riemann Surfaces and Dessins
d'Enfants, London Mathematical Society Student Texts 79, Cambridge University Press, 2012,
pp. 125–126 (the loops around
0,1and∞of the thrice-punctured sphere).
The circle |z| = 3 and the segment joining it to the basepoint #
The point p₊ = 1/2 + (√35/2)·i, where the circle |z| = 3 meets the line re z = 1/2 in the
upper half-plane.
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The vertical segment α₊ from the basepoint 1/2 up to p₊ = 1/2 + (√35/2)·i.
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The circle |z| = 3, traversed counterclockwise once from p₊:
t ↦ 3·exp(i(arccos(1/6) + 2πt)), where p₊ = 3·exp(i·arccos(1/6)). It separates the punctures
0 and 1 from ∞.
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The loop δ lies on the circle of radius 3 about 0, which bounds a punctured disc about
∞ containing neither 0 nor 1.
The closed upper and lower half-planes #
The cut points #
The pieces and the half-planes containing them #
The loop at infinity #
The big circle is the product of the two small ones. The circle |z| = 3, traversed
counterclockwise and transported to the basepoint 1/2 along the vertical segment α₊, is
homotopic in ℂ ∖ {0, 1} to the loop γ0 around 0 followed by the loop γ1 around 1.
periph1 * periph0, the class of γ0 followed by γ1, is the class of the circle |z| = 3
traversed counterclockwise and transported to the basepoint along α₊.
The peripheral element at infinity is the loop around infinity. periphInf is the class of
the circle |z| = 3 traversed clockwise in the affine coordinate z, transported to the basepoint
along α₊. In the chart w = 1/z at ∞ this circle runs counterclockwise about w = 0.
The anharmonic map exchanging 1 and ∞ #
The self-homeomorphism mob1Inf : z ↦ z / (z − 1) fixes the puncture 0, exchanges the punctures
1 and ∞, and moves the basepoint 1/2 to −1. It has real coefficients and reverses the sign
of the imaginary part, so it exchanges the closed upper and lower half-planes, and the half-plane
decomposition of the previous section computes the images of the peripheral loops as well.
The path α₋₁ from −1 = mob1Inf (1/2) to the basepoint 1/2 through the closed upper
half-plane: along the real axis to −1/2, then along the upper half of the circle |z| = 1/2
(range_αMob1Inf). Any two such paths are homotopic, the closed upper half-plane of ℂ ∖ {0, 1}
being simply connected.
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The automorphism of π₁(ℂ ∖ {0, 1}, 1/2) induced by z ↦ z / (z − 1): the isomorphism onto
π₁(ℂ ∖ {0, 1}, −1) induced by the map, followed by the change of basepoint back to 1/2 along
α₋₁. It sends the class of a loop γ to the class of α₋₁⁻¹ ⬝ (mob1Inf ∘ γ) ⬝ α₋₁.
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mob1InfMulAut is the isomorphism induced by z ↦ z / (z − 1) followed by the change of
basepoint along α₋₁.
z ↦ z / (z − 1) fixes the peripheral element at 0.
z ↦ z / (z − 1) carries the peripheral element at 1 to the peripheral element at ∞.
The image of the loop γ1 around 1 is a loop around ∞, and transported back to the basepoint
along α₋₁ its class is periphInf.
The inverse of mob1InfMulAut fixes periph0.
The inverse of mob1InfMulAut carries periph1 to periph1⁻¹ * periphInf * periph1, the
second component of the branch-point operation exchanging 1 and ∞.