manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
A Lagrangian cobordism is roughly a cobordism between Lagrangian submanifolds which is itself a Lagrangian submanifold in a suitable sense.
More in detail, given an ambient symplectic manifold $(X, \omega)$, a Lagrangian cobordism in $X$ is a cobordism with an embedding into $([0,1] \times \mathbb{R} \times X )$ as a Lagrangian submanifold (where $[0,1] \times \mathbb{R} \hookrightarrow \mathbb{R}^2$ is equipped with its canonical symplectic structure) and such that the boundary components are Lagrangian submanifolds of $X$. (for details see e.g. Biran-Cornea 11, 2.1.1).
Lagrangian cobordisms in $(X,\omega)$ arrange into a stable (infinity,1)-category which is supposed to be at least closely related to the Fukaya category of $(X,\omega)$ (Nadler-Tannaka 11).
The notion was introduced in
Further developments are in
Paul Biran, Octav Cornea, Lagrangian cobordism I, (arxiv/1109.4984)
Paul Biran, Octav Cornea, Lagrangian cobordism III(arxiv/1304.6032)
A stable $(\infty,1)$-category of Lagrangian cobordisms is discussed in
David Nadler, Hiro Tannaka, A stable infinity-category of Lagrangian cobordisms, Adv. Math. 366 3 (2020) 107026 [arXiv:1109.4835, doi:10.1016/j.aim.2020.107026]
Wenyuan Li, Lagrangian cobordism functor in microlocal sheaf theory I, arXiv:2108.10914
Last revised on August 7, 2023 at 12:16:53. See the history of this page for a list of all contributions to it.