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I came across a new type of gluing in the FOOO paper (maybe not so new to experts). The situation is as follows: one considers a relative (to lagrangian) class and want to understand the evaluation maps of the moduli space to the lagrangian.

An option is to do the original FOOO thing--Kuranishi structure, multisection perturbation and evaluation; however, if there is a separating contact type hypersurface in the manifold, where the lagrangian entirely lies on one of the pieces, there's another option. I can first do neck-stretching in SFT, then evaluate the fiber product of moduli space involving punctures on the two sides, fiber product taken over evaluation on the punctures where the moduli spaces can be glued. My question is:

When do the two options give the same chain (in an appropriate sense that I don't know how to formulate)$/$class?

I suppose this has been used for a while in GW theory, but I am not sure what can be claimed "standard" in this case of Floer homology. According to the above paper,

When the puncture is simple and of minimal period, this conclusion is standard.

I suspect there's another implicit assumption they used but suppressed here, that the evaluation always gives a cycle due to torus action. I do not have a clear idea why the assumption on simple puncture plays a role. The Li-Ruan's paper and settings therein do not seem to be responsible for this assumption. Notice that, in the setting of the above paper, the evaluation should involve a virtual perturbation since the moduli space in question is (a maslov=$2$ disk)+($k$ copies of maslov=$0$ spheres).

The following special case should help concretize my question: consider still a toric manifold, where the Kuranishi structure and perturbation seem much easier to construct according to toric FOOO I&II. If one considers a certain contact type hypersurface which is Morse-Bott and invariant under torus action, and an arbitrary classes with maslov=$2$, can one still claim such a gluing result "standard"?

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Please, replace the TeX code for your questions with the > code: see the "how to ask" section. – Marco Golla Jul 3 '11 at 16:51
Thanks Marco for pointing this out! – Weiwei Jul 3 '11 at 18:03
Tim, you are correct. I deleted the comment. – Orbicular Jul 4 '11 at 8:58
Certainly I'm not an expert on any of this but at least to compute Floer homology or some A(\infty) product and perturb your Lagrangians. If you had good transversality properties and only care about 0-dimensional moduli spaces than the argument that for large values of your stretch parameter the "fibre product" moduli space agrees with the "actual" moduli space seems standard to the untrained eye? As for K. structures, where it's already hard for many experts to tell whether all the details are there, add SFT and general sym. cobordism well that sounds like a nightmare doesn't it? – Daniel Pomerleano Jul 4 '11 at 12:36
To Daniel: one scary part of the FOOO story is the J-dependence of the $A_\infty$ structure. I haven't been patient/brave enough to look into the particular sections in their book, but it is a priori different from the usual story. Homotopy in J only provides a certain homomorphism on the underlying algebra/module, to claim it is an isomorphism, the easiest and only way off-hand in the literature that I know of is to assume the genericity of this whole path of J. – Weiwei Jul 4 '11 at 16:12

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