# Pseudocycle definition of open Gromov-Witten invariants

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as long!)

I'm taking the following set-up directly from page 29 of http://www-math.mit.edu/~auroux/papers/slagmirror.pdf

Let $(M,\omega, J)$ denote a compact symplectic manifold and compatible almost complex structure, and let $L$ be a compact, oriented Lagrangian submanifold of $M$.

Given a class $\beta \in \pi_2 (X, L)$, denote by $\mathcal{M}(L, \beta)$ the space of parameterized $J$-holomorphic maps from $(D , \partial D )$ to $(X, L)$ representing the class $\beta$ (parameterized means we don't quotient by automorphisms of the disc).

For "marked points" $\pm 1 \in \partial D$, and $0\in D$, let $$ev_{\beta,\pm 1} : \mathcal{M}(L, \beta)\to L$$ and $$ev_{\beta,0} : \mathcal{M}(L, \beta)\to M$$ denote the corresponding evaluation maps (sending a $J$-holomorphic disk to its image at the point in question).

Auroux says that up to introducing suitable perturbations (of the Cauchy Riemann equations, I presume) we can assume $\mathcal{M}(L, \beta)$ carries a fundamental chain, and that the evaluation maps can be chosen to be transverse to fixed chains in $L$ and $X$.

QUESTION: Is there any detailed write up of the above claim? (The new work by FOOO?) I'm actually interested in a slightly more general case, where we consider moduli spaces of disks between two different Lagrangians, but I assume the same arguments will work in both cases (given proper assumptions on the Lagrangians and symplectic manifolds in question). Any reference would be appreciated (with any caveats about the reference, if relevant).

EDIT: let me stress that I'm interested in the simplest cases, eg monotone symplectic manifolds with monotone Lagrangians.

Note: Many of the applications of the above constructions can be accomplished with an alternative construction, which examines moduli spaces of disks with extra $S^1$ boundary components, required to lie on generic orbits of a generic Hamiltonian vector field, rather than basepoints required to lie on chains (which is the above method). I think this second approach is technically easier, but I haven't been able to fit what I need into it, unfortunately. But I do have a related QUESTION about this approach: as far as I can see, it is only a substitute to the chain-level approach in the case of interior marked points. Is there an analogous construction providing an alternative to boundary marked points?

Thanks!

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In the notes you're citing, the class $\beta$ has the minimal amount of energy among classes admitting holomorphic discs, so there is no bubbling issues. In particular, the moduli space is, for generic almost complex structure, a smooth manifold. The fact that evaluation maps at interior points can be made transverse to any cycle in $M$ can be extracted from work of Floer-Hofer-Salamon. For cycles on Lagrangians, you can use a doubling trick. –  Mohammed Abouzaid Dec 31 '11 at 1:29
For the compact case, what you are looking for is probably Proposition 3.4.5 (or 7.2.35) in FOOO's book. But situation is considerably complicated there, e.g. you need a system of abtract perturbations instead of just one to make everything work, which forces one to choose a countable subcomplex of singular (co)chains. –  Weiwei Jan 1 '12 at 18:34
I never said thanks for your answers! Because I went away to try to understand them. I think I do now. Thanks! –  Sam Lewallen Jan 10 '12 at 5:47