nLab
Lagrangian cobordism

Contents

Context

Manifolds and cobordisms

Symplectic geometry

Contents

Idea

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,ω)(X, \omega), a Lagrangian cobordism in XX is a cobordism with an embedding into ([0,1]××X)([0,1] \times \mathbb{R} \times X ) as a Lagrangian submanifold (where [0,1]× 2[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 XX. (for details see e.g. Biran-Cornea 11, 2.1.1).

Lagrangian cobordisms in (X,ω)(X,\omega) arrange into a stable (infinity,1)-category which is supposed to be at least closely related to the Fukaya category of (X,ω)(X,\omega) (Nadler-Tannaka 11).

References

The notion was introduced in

Further developments are in

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

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