Hi all,


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 


1. Is this anywhere near correct, and if so, has this been done before?
2. 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?



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.


