Flux homomorphism for manifolds with boundary

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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?
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1
Shouldn't your map be circle-valued? The ambiguity is the area of the annulus. – Tim Perutz Feb 10 2010 at 0:46
Thanks for pointing this out. It took me a while to get it, but you're right: when choosing the 2-cycle S you can add any multiple of the fundamental class. I guess I need to add the condition that $[S] = 0$ as an element of $H_2$. Hmm, this is a good point -- I'm not sure how this is addressed in the usual definition of the flux homomorphism. – jvkersch Feb 11 2010 at 12:11
Of course, in the usual setup the flux homomorphism is circle valued -- that's the whole point. Sorry for being so slow on the uptake, I realized it the moment I added my comment. – jvkersch Feb 11 2010 at 12:13

1 Answer

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.

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1) You're absolutely right: the other "generator" that I mentioned in my question is of course homologous to the boundary, hence zero.  This also puts an end to much of my confusion about relative homology, so thanks again!  2-3) You're right about volume vs. symplectic manifolds.  I'm trying to formalize some construction in Arnold's book on Topological Fluid Dynamics, so luckily volume-preserving maps are what I need.  Putting it in the context of manifolds with a volume form gives me a pointer of where to look for references -- I remember seeing a flux homomorphism for volume manifolds... – jvkersch Feb 10 2010 at 10:19
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