Lagrangian surgery

Lagrangian surgery replaces two transverse local Lagrangian sheets by a smooth Lagrangian neck. For the standard planes ${\mathbb{R}}^n$ and $i{\mathbb{R}}^n$ in ${\mathbb{C}}^n$, a neck is obtained by following a curve between the two coordinate rays and multiplying it by points of $S^{n-1}$. In dimension four, resolving a double point attaches a one-handle to the source surface and lowers its Euler characteristic by two.

If two embedded Lagrangian surfaces meet transversely in one point, Lagrangian surgery at that point produces their connected sum. For orientable surfaces ${\Sigma }_{g_1}$ and ${\Sigma }_{g_2}$, the result is ${\Sigma }_{g_1+g_2}$.

Perturb a clean self-intersection circle of a Lagrangian immersion by a Morse function on $S^1$. Its minimum and maximum give two transverse double points. Resolving both by Lagrangian surgery adds two one-handles and changes the Euler characteristic by $-4$. One neck may be glued with the orientation-reversing choice, so the resolved surface is nonorientable.

For every $l\ge 1$, take a plane immersed circle with $l$ transverse double points and form its product with an embedded circle in a second symplectic plane. The resulting torus is a Product Lagrangian immersion with $l$ clean double circles. Resolving them gives a connected nonorientable Lagrangian embedding in ${\mathbb{R}}^4$ with Euler characteristic $-4l$.