By our assumption that c 0 and c 1 are piecewise smooth homotopic, there is a piecewise smooth homotopy H : D → M Γ Γ i ( t ) = H ( γ γ i ( t ) ) i = 1 , 2 , 3 , 4 Γ Γ ( t ) = H ( γ γ ( t ) ) = ( Γ Γ 1 ⊕ ⊕ Γ Γ 2 ⊕ ⊕ Γ Γ 3 ⊕ ⊕ Γ Γ 4 ) ( t )