Relative symplectic-area independence

Two homotopy strips with the same endpoint paths and boundary on Lagrangian submanifolds glue to a relative $2$-cycle. If that cycle is zero in $H_2(M,L_0\cup L_1)$, its symplectic area vanishes: apply Stokes theorem, using $d\omega =0$ and the zero restriction of $\omega$ to each Lagrangian. Thus relative homology controls the ambiguity in a Lagrangian path-area functional.