Manuscript VII-F@1 · VII-F.tex · section · 2d81c03f

Holonomy through the changed particle star

Adapted affine charts and exact holonomy › Holonomy through the changed particle star · lines 1211–1243

The preceding common meridian provides a certificate for the whole window. The new star itself has the same nontrivial peripheral word.

For $s > 0$ , the loop $ρ = s$ lies strictly inside the six-face star, because

h - k \leq h = \frac{2}{3} s < s .

In angular order its charts are

U_{0}^{-}, U_{0}^{+}, U_{1}^{-}, U_{1}^{+}, U_{2}^{-}, U_{2}^{+} .

Let $A_{0}, \dots, A_{5}$ denote these six adapted coordinate changes. The ordered product is

(A_{0} J A_{5}^{- 1}) (A_{5} A_{4}^{- 1}) \dots (A_{1} A_{0}^{- 1}) = A_{0} J A_{0}^{- 1} .

Transporting the framing to the common collar again gives $J$ .

For $s < 0$ , a sufficiently small meridian meets the three $V_{i}$ faces and has the three-factor product already computed. The local outgoing loop may shrink as $s ↓ 0$ ; it is not used as the nonvanishing loop at the event parameter. Its correspondence with $γ_{s}$ is the explicit annular homotopy in the punctured slice.

Displayed formulas in this passage

unlabeled · line 1218
$h - k \leq h = \frac{2}{3} s < s .$
unlabeled · line 1222
$U_{0}^{-}, U_{0}^{+}, U_{1}^{-}, U_{1}^{+}, U_{2}^{-}, U_{2}^{+} .$
unlabeled · line 1227
$(A_{0} J A_{5}^{- 1}) (A_{5} A_{4}^{- 1}) \dots (A_{1} A_{0}^{- 1}) = A_{0} J A_{0}^{- 1} .$