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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