Lemma 9 ([Moving the graph map off the protected discs]). Suppose there are pairwise disjoint proper locally flat embeddings f_r:D^2⟶M with f_r|_ S^1 =f_r|_ S^1 . Then p_0 is homotopic, relative to ∂G , to a map p:G⟶Γ with the following …

Join the boundary of each actual embedded disc back to α_r by the radial annulus in the deleted base tube. The bases and these return annuli lie in V and hence already map into Γ_h . The boundary map of each disc lies in Γ_h and is nullhomotopic in Γ because the actual disc gives a filling. The inclusion π_1(Γ_h)⟶π_1(Γ) is injective, so it has a filling in Γ_h . The original filling and this new filling are homotopic relative boundary since π_2(Γ)=0 . Transfer this homotopy to the disc image using the embedding parametrization. This step uses no homotopy between an output disc and its input cap. Let B be the union of the bases, return annuli, and disc images, and put A=B∪∂G . Prescribe the just constructed homotopies on the discs and keep the other parts fixed. These prescriptions agree on intersections: disc boundaries are fixed, disc interiors are mutually disjoint and proper in M , and the only intersections with the returned pieces are the stated attaching circles. Pasting finitely many compact closed sets gives a continuous homotopy on A . Only B is to map into Γ_h ; the q -collar portions of ∂G keep their prescribed product values. Here the extension to G need not assume that the topological disc images are smooth or that their inclusion is a cofibration. A finite graph is an absolute neighborhood retract for metric spaces. Apply its neighborhood-extension property to the prescribed map on the closed subset (G×0)∪(A×[0,1]) ⊂G×[0,1]. It extends to an open neighborhood O of that subset. Compactness of A×[0,1] gives a neighborhood U of A with U×[0,1]⊂O . Choose a continuous function χ:G⟶[0,1] equal to one on A and supported in U . If H is the neighborhood extension, then H(x,t)=H(x,χ(x)t) is defined on all of G×[0,1] : when χ(x)>0 it uses U×[0,1] , and otherwise it uses the time-zero slice. It is the required extension, fixed on ∂G . Its final map sends B into Γ_h . Choose mutually disjoint small open intervals about the midpoints of the q_e edges. Their closed smaller intervals miss Γ_h , so their inverse images miss the protected compact set B . On ∂G the map is already the fixed coordinate on the single collar C_e=T^2×[0,1] . In a small boundary collar straighten its real interval coordinate to be constant in the inward normal direction, using an interpolation fixed on the boundary. Use a slightly larger target interval to localize this operation away from B . Relative smooth approximation of real functions, followed by relative transversality in the remaining interior, makes the map smooth and transverse near a smaller middle level, preserving the boundary product and staying off B . The intervals for different edges have disjoint supports. These changes are homotopies within intervals, so the resulting p is still homotopic to p_0 relative boundary. The regular-value and boundary regular-fiber statements are those of ; the fixed product collar supplies regularity on the boundary. Each level J_e is therefore a compact smooth hypersurface, cooriented by the interval and oriented by G . Its boundary is exactly the boundary level T^2×point in C_e ; Q contributes no boundary point at that level. There is one component with this boundary torus, and every other component is closed. All the topological disc images and returned pieces remain disjoint from the cuts.