I am wondering whether someone has considered the definition of the flux homomorphism for manifolds with boundary. More specifically, I am looking at the annulus and I want the diffeomorphism $\phi(r, \theta) = (r, \theta + \alpha)$ for $\alpha$ constant to have non-zero flux.
I was thinking of extending the flux homomorphism using the relative homology like this:
$flux: Symp_0(A) \times H_1(A, \partial A) \to R$
where the value of $flux$ on the pair $(\phi, C)$ is given by choosing a 2-cycle $S$ such that $\partial S = C - \phi(C)$ and integrating the symplectic form over this cycle. Now, I think $H_1(A, \partial A)$ is generated by a circle homologous to the boundary and an arc connecting the two, and if you compute the value of $flux$ on the rotiational diffeomorphism and the latter generator, you get a non-zero result.
My questions are
- Is this anywhere near correct, and if so, has this been done before?
- Is there a way to define the flux homomorphism in exactly the same way as for the case of manifolds without boundary, so that the standard results are still valid?