\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: the Lagrangian surgery exact triangle

the Lagrangian surgery exact triangle

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Now that we have a geometric description of \(L_1\#_\lambda L_2\), we discuss what object it represents in the Fukaya category. We restrict ourselves to \(\dim_\RR(X)=2\) so that we may draw pictures. However, the pictures are only for intuition (and in fact the sketch of proof we give only work when \(\dim(X)\geq 4\)). Let \(L_0\) be some test Lagrangian, which intersects both \(L_1\) and \(L_2\) as in figure 0.0.1. If the surgery neck is chosen to lie in a neighborhood disjoint from the intersections \(L_0\cap (L_1\cup L_2)\), then these intersections are in bijection with the intersections \(L_0\cap (L_1\# L_2)\). Therefore \(\CF(L_0, L_1\# L_2)=\CF(L_0, L_1)[1]\oplus \CF(L_, L_2)\) as vector spaces.
figure 0.0.1:By rounding the corner, we can compare holomorphic triangles with holomorphic strips on the surgery.
The intuition from [FOOO07] is that there is a bijection between certain holomorphic triangles with boundary on \(L_0, L_1, L_2\) which passes through the intersection point \(p_{12}\), and holomorphic strips with boundary on \(L_0\) and \(L_1\# L_2\). Since holomorphic triangles contribute to the \(\emprod^3\) structure coefficients, and strips to the differential, it is reasonable to hope that we can state a relation between \(L_1, L_2,\) and \(L_1\#L_2\) as objects of the Fukaya category. First, we observe that the intersection point \(p_{12}\) determines a morphism in \(\hom(L_2, L_1)\). Since we've assumed that \(L_1\) and \(L_2\) intersect at only one point, we know that \(\emprod^1(p_{12})=0\). We can therefore form the twisted complex \(\cone(p_{12})\). We now provide justification for why this is isomorphic to \(L_1\# L_2\). We have already observed that for our test Lagrangian \(L_0\) we had an isomorphism of vector spaces between \(\hom(L_0, L_1)\oplus \hom(L_0, L_2)\) and \(\hom(L_0, L_1\#_\lambda L_2)\). The differential on \(\hom(L_0, L_1\#_\lambda L_2)\) comes from counting holomorphic strips, which we break into two types: those which avoid a neighborhood of the surgery neck, and those which pass through the surgery neck.

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 \(L_0, L_1\#_\lambda L_2\). The more difficult portion is to understand the strips which pass through the neck.

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 In fact, [FOOO07] proves the above statement in much greater generality than we state here. The condition that \(\dim(X)\geq 4\) can already be seen in in figure 0.0.1. Observe that if we have a pseudoholomorphic triangle whose boundary passes through the point \(p_{12}\) in the wrong way, that there is no corresponding pseudoholomorphic strip with boundary on \(L_0\) and \(L_1\# L_2\). Ignoring the potential complications in the definition of the Fukaya category, we obtain: Let \(\dim(X)\geq 4\), and \(L_1, L_2\) be exact Lagrangian submanifolds which intersect transversely at a single point \(p_{12}\). Then \(L_1\#_{\lambda} L_2\) is isomorphic to the twisted complex \((L_1[1]\oplus L_2, \emprod^2(T^{-\lambda} p_{12}-))\) in \(\Fuk(X)\).

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.