Lagrangian cobordism

**manifolds** and **cobordisms**

cobordism theory, *Introduction*

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

- Vladimir Arnold,
*Lagrange and Legendre cobordisms. I, II*, Funkts. Anal. Prilozh. 14:3, 1–13 (1980) article; 14:4 (1980), 8–17 article

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 (infinity,1)-category of Lagrangian cobordisms is discussed in

- David Nadler, Hiro Tannaka,
*A stable infinity-category of Lagrangian cobordisms*(arXiv:1109.4835)

Last revised on November 10, 2013 at 10:52:11. See the history of this page for a list of all contributions to it.