Lagrangian -surgery modifies an immersed Lagrangian submanifold by topological -surgery while removing a self-intersection. Associated to a -surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. This exact homotopy can be chosen with as small Hofer norm as desired. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated with the handle attachment to the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds. In an appendix, we describe a 2-ended embedded monotone Lagrangian cobordism which is not the suspension of a Hamiltonian isotopy following a suggestion of Abouzaid and Auroux.
Cite this article
Jeff Hicks, Lagrangian cobordisms and Lagrangian surgery. Comment. Math. Helv. 98 (2023), no. 3, pp. 509–595DOI 10.4171/CMH/554