proposition 0.0.2
Let \(\dim_\RR(X)\geq 4\), and let \(L_1, L_2\) be exact Lagrangian submanifolds which intersect at a single point \(p_{12}\). then we can choose an almost complex structure \(J\) so that whenever \(p_{01}, q_{01}\in L_0\cap L_1\) are intersections, and \(u: [0,1]\times \RR\to X\) is a \(J\) holomorphic strip, then the boundary of \(u\) is disjoint from a small neighborhood of \(L_1\cap L_2\). In particular, \(u\) gives a \(J\)-holomorphic strip with boundary on \(L_0, L_1\#_\lambda L_2\).theorem 0.0.3 [FOOO07]
Let \(\dim(X)\geq 4\), and \(L_1, L_2\) be exact Lagrangian submanifolds which intersect transversely at a single point \(p_{12}\). Let \(L_0\) be another exact Lagrangian submanifold which intersects \(L_1, L_2\) transversely. Then for sufficiently small surgery necks, there exists a choices of almost complex structure on \(X\) for which we have a bijection between- \(J\)-holomorphic strips with boundary on \(L_0, L_1\# L_2\) which pass through the surgery neck;
- \(J\)-holomorphic triangles with boundary on \(L_0, L_1, L_2\).
References
[FOOO07] | K Fukaya, YG Oh, H Ohta, and K Ono. Lagrangian intersection Floer theory-anomaly and obstruction, chapter 10. Preprint, can be found at http://www. math. kyoto-u. ac. jp/˜ fukaya/Chapter10071117. pdf, 2007. |