Biran and Cornea showed that monotone Lagrangian cobordisms give an equivalence
of objects in the Fukaya category. However, there are currently no known nontrivial
examples of monotone Lagrangian cobordisms with two ends. We look at an
extension of their theory to the pearly model of Lagrangian Floer cohomology and
unobstructed Lagrangian cobordisms. In particular, we examine the suspension
cobordism of a Hamiltonian isotopy and the Haug mutation cobordism between
mutant Lagrangian surfaces. In both cases we show that these Lagrangian
cobordisms can be unobstructed by a bounding cochain, and additionally induce an
homomorphism between the Floer cohomology of the ends. This gives a first example
of a two-ended Lagrangian cobordism giving a nontrivial equivalence of Lagrangian
Floer cohomology.
A brief computation is also included which shows that the incorporation of a
bounding cochain from this equivalence accounts for the “instanton-corrections”
considered by Auroux (2007), Pascaleff and Tonkonog (2020) and Rizell, Ekholm and
Tonkonog (2022) for the wall-crossing formula between Chekanov and product tori in
.
We additionally prove some auxiliary results that may be of independent
interest. These include a weakly filtered version of the Whitehead theorem for
algebras and an extension of Charest and Woodward’s stabilizing divisor model of
Lagrangian Floer cohomology to Lagrangian cobordisms.
