Manuscript VIII-A@3 · VIII-A-v3.tex · lemma · 2398a747

Lemma: Canonical word formula

Holonomy on the marked punctured disk › Ordered products and based transport · lines 1165–1189 · proof included or linked

VIII-A@3#A:lem:word, VIII-A@3#A:eq:word

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For a closed canonical chart path,

Hol (γ) = A_{f_{0}} J (μ)^{N (γ)} A_{f_{0}}^{- 1}, N (γ) = k = 0 \sum^{m - 1} n_{f_{k + 1} f_{k}} .

On the marked punctured cone slice, $N (γ)$ is its oriented winding number.

Proof. Substitute A:eq:canonical. Adjacent adapted-chart factors cancel, leaving the displayed power of $J (μ)$ .

Lift angular position to the cover with period $L_{μ} = 2 π - μ$ . A positive crossing of the marked seam increases the sheet index by one; a negative crossing decreases it by one. A closed path has angular increment $N (γ) L_{μ}$ , which is its winding number times the angular period.

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VIII-A@3#A:eq:word · line 1168
$Hol (γ) = A_{f_{0}} J (μ)^{N (γ)} A_{f_{0}}^{- 1}, N (γ) = k = 0 \sum^{m - 1} n_{f_{k + 1} f_{k}} .$

Lemma: Canonical word formula

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