  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.
