Volume 19, issue 2 (2019)

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Topology of (small) Lagrangian cobordisms

Mads R Bisgaard

Algebraic & Geometric Topology 19 (2019) 701–742

We study the following quantitative phenomenon in symplectic topology: in many situations, if a Lagrangian cobordism is sufficiently small (in a sense we specify) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms, which, by Biran and Cornea’s Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev’s theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.

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Lagrangian cobordisms, Floer homology, Lagrangian submanifolds
Mathematical Subject Classification 2010
Primary: 53D05, 53D12, 53D40
Received: 8 August 2017
Revised: 26 June 2018
Accepted: 14 August 2018
Published: 12 March 2019
Mads R Bisgaard
Department of Mathematics
ETH Zürich