Context
Replacing labeled boundary circles by marked points forgets full twists around those circles. The quotient alone therefore cannot reconstruct a mapping class.
Definitions
- π is the capping map from boundary-pointwise mapping classes to mapping classes fixing the new marked points.
- D=(D₁,…,D_b) is normalized by D(t_{δ_j})=e_j on interior boundary twists.
Hypotheses and scope
- The ambient surface is a disk.
- Interior boundary components and the resulting marked points are labeled and fixed individually.
Proof or evidence
The page states the kernel of π and the normalization of D used by every subsequent theorem.
Verification notes
The labeled-boundary and pointwise-fixing hypotheses were checked against the thesis conventions; the unlabeled and closed-surface cases are explicitly excluded.
Limitations
- If boundaries may permute, the coordinates transform and the product becomes semidirect rather than direct.
- This definition alone does not classify positive factorizations.