The power-map charts of the components over the punctures #
Let p : E → ℂ ∖ {0, 1} be a covering map whose fibre over the basepoint 1/2 is numbered by
ν : p ⁻¹' {1/2} ≃ Fin n, with monodromy triple (σ0, σ1, σinf). For each puncture
q ∈ {0, 1, ∞} the connected components of the preimage p ⁻¹' D_q* of the standard punctured
neighborhood
D₀* = {z | ‖z‖ < 1/2},D₁* = {z | ‖z - 1‖ < 1/2},D∞* = {z | 2 < ‖z‖}
are the cycles of σ_q (IsCoveringMap.sameCycleQuotientσ0EquivConnectedComponents and its
analogues at 1 and ∞). This file computes the degree of each of them, and with it the local
model of the cover over D_q*:
- the component of the cycle of the sheet
ihas as many points over the local basepoint (zeroBasePt,oneBasePt,pPlus) as the cycle ofiunderσ_qhas elements, namelyFunction.minimalPeriod σ_q i; - in the coordinates
z ↦ 2 * z,z ↦ 2 * (1 - z)andz ↦ 2 / z, which identifyD₀*,D₁*andD∞*with the punctured unit disc𝔻*, the restriction ofpto that component is isomorphic over𝔻*to the power mapw ↦ w ^ eexactly wheneis the length of the cycle ofi.
These isomorphisms are the charts along which the puncture of each component is filled in when
the cover is compactified to a branched cover of the Riemann sphere: the added point is w = 0,
and the compactified cover is w ↦ w ^ e near it, with e the length of the cycle. Each chart is
unique only up to a rotation by an e-th root of unity
(TauCeti.existsUnique_rootsOfUnity_smul_homeomorph).
The three punctures share one argument: the lifts of the path α_q from 1/2 into D_q* number
the points of the fibre over the local basepoint, and two of them lie in one component exactly
when their sheets lie in one cycle of σ_q
(IsCoveringMap.sameCycle_monodromyTriple_σ0_iff_joinedIn and its analogues).
Main declarations #
IsCoveringMap.ncard_connectedComponent_inter_fiber_σ0,IsCoveringMap.ncard_connectedComponent_inter_fiber_σ1,IsCoveringMap.ncard_connectedComponent_inter_fiber_σinf: the component of the cycle ofihasFunction.minimalPeriod σ_q ipoints over the local basepoint.IsCoveringMap.exists_homeomorph_puncturedDiscPow_iff_minimalPeriod_σ0,IsCoveringMap.exists_homeomorph_puncturedDiscPow_iff_minimalPeriod_σ1,IsCoveringMap.exists_homeomorph_puncturedDiscPow_iff_minimalPeriod_σinf: the restriction ofpto that component is, in the coordinate ofD_q*, isomorphic tow ↦ w ^ eexactly whene = Function.minimalPeriod σ_q i.
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, §1.2.7 (the components of the preimage of a punctured disc and their power-map charts) and §2.7 (the local degrees are the cycle lengths of the monodromy).
- O. Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81, Springer 1981, §5, Theorem 5.10, and §8, Theorem 8.4.
Degrees and charts from the cycles of a local monodromy #
The puncture 0 #
The degree of a component over 0 is the length of its cycle. The connected component of
p ⁻¹' D₀* that corresponds to the cycle of the sheet i under
IsCoveringMap.sameCycleQuotientσ0EquivConnectedComponents contains exactly
Function.minimalPeriod σ0 i points over zeroBasePt = 1/4.
The components over 0 are power maps. In the coordinate z ↦ 2 * z, which identifies
the punctured neighborhood D₀* of 0 with the punctured unit disc 𝔻*, the restriction of p
to the component of p ⁻¹' D₀* corresponding to the cycle of the sheet i is isomorphic over
𝔻* to the power map w ↦ w ^ e, for e ≠ 0, exactly when e is the length
Function.minimalPeriod σ0 i of that cycle.
The puncture 1 #
The degree of a component over 1 is the length of its cycle. The connected component of
p ⁻¹' D₁* that corresponds to the cycle of the sheet i under
IsCoveringMap.sameCycleQuotientσ1EquivConnectedComponents contains exactly
Function.minimalPeriod σ1 i points over oneBasePt = 3/4.
The components over 1 are power maps. In the coordinate z ↦ 2 * (1 - z), which
identifies the punctured neighborhood D₁* of 1 with the punctured unit disc 𝔻*, the
restriction of p to the component of p ⁻¹' D₁* corresponding to the cycle of the sheet i is
isomorphic over 𝔻* to the power map w ↦ w ^ e, for e ≠ 0, exactly when e is the length
Function.minimalPeriod σ1 i of that cycle.
The puncture ∞ #
The degree of a component over infinity is the length of its cycle. The connected
component of p ⁻¹' D∞* that corresponds to the cycle of the sheet i under
IsCoveringMap.sameCycleQuotientσinfEquivConnectedComponents contains exactly
Function.minimalPeriod σinf i points over pPlus: the endpoints of the lifts of αPlus
starting at the sheets of that cycle.
The components over infinity are power maps. In the coordinate z ↦ 2 / z, which
identifies the punctured neighborhood D∞* of infinity with the punctured unit disc 𝔻*, the
restriction of p to the component of p ⁻¹' D∞* corresponding to the cycle of the sheet i is
isomorphic over 𝔻* to the power map w ↦ w ^ e, for e ≠ 0, exactly when e is the length
Function.minimalPeriod σinf i of that cycle.