dg.differential geometry - Flux homomorphism for manifolds with boundary

Here are some remarks about your definition.

1) $H_1(A,partial A)$ is just one-dimensional, it is generated by a path that joins two sides of $A$.

2) The definition that you gave works for the annulus, and for surfaces with a boundary as well. This will also work for manifolds with boundary $M^n$, in the case
when you consider fluxes of volume-preserving maps (i.e. you work with $Omega^n$ and $H_{n-1} (M^n, partial M^n$). I have not seen this definition before, but it is so natural, that it would be strange if no one considered it.

3) It does not look that this definition will work for higher-dimensional symplectic manfiolds, if you want to study fluxes of symplectomorphisms (and you work with $omega$ and $H_1(M^{2n},partial M^{2n})$), because the restriction of $omega$ to $partial$ will be non-zero.