proposition 0.0.2 [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 0.0.4
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 0.0.5
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 0.0.6. 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 \(S^1\).References
[Pol91] | Leonid Polterovich. The surgery of Lagrange submanifolds. Geometric & Functional Analysis GAFA, 1(2):198--210, 1991. |