Lagrangian cobordisms and Lagrangian surgery

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

Lagrangian k-surgery modifies an immersed Lagrangian submanifold by topological k-surgery while removing a self-intersection. 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. 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.
Original languageEnglish
Pages (from-to)509-595
JournalCommentarii Mathematici Helvetici: A Journal of the Swiss Mathematical Society (CMH)
Volume98
Issue number3
DOIs
Publication statusPublished - 4 Nov 2023

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