The definition implies that each defect line δ of Σ corresponds either to a single defect line of Σ ′ with the same label as δ , or else splits into several defect lines of Σ ′ that all carry the same label as δ and which are interrupted by fillable gluing circles that are not present in Σ. Moreover, the gluing boundaries of Σ correspond to identical gluing boundaries of Σ ′ . As an illustration, the following picture shows the refinement ( D ; D ′ ) of a one-holed disk D for which D ′ has three additional gluing circles: D = D ′ = (5.26) Remark 5.13. (i) For any refinement (Σ; Σ ref ), by definition every gluing boundary of Σ ref that is not a gluing boundary of Σ is fillable. (ii) It is worth comparing the notion of refinement with Definition 5.10 of a fillable disk of type X : It is easily seen that for every refinement ( X , X ref ) with X a defect surface for which the underlying surface X is a disk and which has at most one free boundary segment, the
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- Fall '17