Invariance of immersed Floer cohomology under Lagrangian surgery

Abstract

We show that Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The chain-level map replaces the two orderings of the self-intersection point with meridional and longitudinal cells on the handle created by the surgery and uses a bijection between holomorphic disks developed by Chapter 10 of Fukaya–Oh–Ohta–Ono (2009). Our result generalizes the invariance of potentials for certain Lagrangian surfaces in Theorem 1.2 of Dimitroglou Rizell–Ekholm–Tonkonog (2022) and implies the invariance of Floer cohomology under mean curvature flow with this type of surgery, as conjectured by Joyce (2015).