Lagrangian cobordisms and Lagrangian surgery

Abstract

Lagrangian surgery and Lagrangian cobordism give geometric interpretations to exact triangles in Floer cohomology. Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection point of the immersion. Associated to a $k$-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. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated to the handle attachment with 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, recover the Lagrangian surgery exact sequence, and provide conditions for when non-monotone Lagrangian cobordisms yield continuation maps in the Fukaya category.