1: Lagrangian connect sum
We will now look at some ways to glue together new Lagrangian submanifolds from old. A source of inspiration for us will be from smooth topology, where tools such as surgery, Morse theory, and cobordisms provide methods for generating new manifolds. An example where we can take a method from topology and directly import it into symplectic geometry is connect sum for Lagrangian curves inside of surfaces. In this setting, two Lagrangian curves which intersect at a point are modified at the point of intersection to produce a connected Lagrangian submanifold. See figure 1.0.1. The Polterovich connect sum is a generalization of this surgery to higher dimensions, which smooths out a transverse intersection between two Lagrangian submanifolds. The idea of construction is to take a standard model for the transverse intersection, then construct a model neck in that canonical neighborhood.proposition 1.0.2 [Pol91]
Let \(L_1, L_2\subset X\) be two Lagrangian submanifolds intersecting transversely at a single point \(p\). Then there exists a Lagrangian submanifold \(L_1\#_p L_2\subset X\) which- topologically is the connect sum of \(L_1\) and \(L_2\) at \(p\).
- Agrees with \(L_1\cup L_2\) outside of a small neighborhood of \(p\) in the sense that \[L_1\#_p L_2|_{X\setminus U}=L_2\cup L_2|_{X\setminus U}.\]
- \(\gamma(t)=t\) for \(t< t_0\), and
- \(\gamma(t)=\jmath t\) for \(t>t_0\).
proposition 1.0.4
Let \(L_0, L_1\) be two Lagrangian submanifolds which intersect transversely at a point, and let \(U\) be a neighborhood of the point in which we implant a surgery neck. Suppose two profile curves \(\gamma_0, \gamma_1\) have the same flux \(\lambda\). Then there exists a Hamiltonian supported on \(U\) whose time one identifies \(L_1\#_{\gamma_0} L_2\) and \(L_1\#_{\gamma_1} L_2\).example 1.0.5
We now visualize the Polterovich surgery for Lagrangian sections of \(T^*\RR^n\). The Lagrangians which we consider are two sections of the cotangent bundle. Let \(L_1\) be the graph of \(d(q_1^2+ \cdots+ q_n^2)\), and let \(L_2\) be the graph of \(d(-q_1^2-\cdots -q_n^2)\). In dimension 2, we can then draw \(L_1\# L_2\) and \(L_2\# L_1\) as multisections of the cotangent bundle. These multisections are sketched in figure 1.0.6. Note that one of surgeries creates a Lagrangian which has a ``neck'' visible in the projection to the base of the cotangent bundle. The other surgery is generically a double-section of the cotangent bundle, except over the fiber of the intersection point where it is instead an \(S^1\).2: the Lagrangian surgery exact triangle
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 2.0.1. 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. 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 2.0.2
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 2.0.3 [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. |
[Pol91] | Leonid Polterovich. The surgery of Lagrange submanifolds. Geometric & Functional Analysis GAFA, 1(2):198--210, 1991. |