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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 10th 2013

Added to Maslov index and to Lagrangian Grassmannian the following quick cohomological definition of the Maslov index:

The first ordinary cohomology of the stable Lagrangian Grassmannian with integer coefficients is isomorphic to the integers

$H^1(LGrass, \mathbb{Z}) \simeq \mathbb{Z} \,.$

The generator of this cohomology group is called the universal Maslov index

$u \in H^1(LGrass, \mathbb{Z}) \,.$

Given a Lagrangian submanifold $Y \hookrightarrow X$ of a symplectic manifold $(X,\omega)$, its tangent bundle is classified by a function

$i \;\colon\; Y \to LGrass \,.$

The _Maslov index of $Y$ is the universal Maslov index pulled back along this map

$i^\ast u \in H^1(Y,\mathbb{Z}) \,.$
• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeSep 18th 2022
• M. V. Finkelberg, Orthogonal Maslov index, Funct. Anal. Appl. 29(1) 72–74 (1995) doi
• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeSep 20th 2022
• Alan Weinstein, The Maslov cycle as a Legendre singularity and projection of a wavefront set, Bull. Braz. Math. Soc., N.S. 44, 593–610 (2013) doi