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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded 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]] $$ H^1(LGrass, \mathbb{Z}) \simeq \mathbb{Z} \,. $$ [[generalized the|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 [[classifying space|classified]] by a function $$ i \;\colon\; Y \to LGrass \,. $$ The _[[Maslov index]] of $Y$ is the universal Maslov index pulled back along this map $$ i^\ast u \in H^1(Y,\mathbb{Z}) \,. $$

   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

   $$i^\ast u \in H^1(Y,\mathbb{Z}) \,.$$
2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 18th 2022
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   Author: zskoda
   Format: MarkdownItex* M. V. Finkelberg, _Orthogonal Maslov index_, Funct. Anal. Appl. 29(1) 72–74 (1995) [doi](https://doi.org/10.1007/bf01077047) <a href="https://ncatlab.org/nlab/revision/diff/Maslov+index/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Maslov+index/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Maslov+index">current</a>

   • M. V. Finkelberg, Orthogonal Maslov index, Funct. Anal. Appl. 29(1) 72–74 (1995) doi

   diff, v13, current

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 20th 2022
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   Author: zskoda
   Format: MarkdownItex* [[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](https://doi.org/10.1007/s00574-013-0026-6) <a href="https://ncatlab.org/nlab/revision/diff/Maslov+index/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/Maslov+index/14">v14</a>, <a href="https://ncatlab.org/nlab/show/Maslov+index">current</a>

   • 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

   diff, v14, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 4th 2024
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   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Paolo Piccione]], [[Alessandro Portaluri]], [[Daniel V. Tausk]]: *Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics*, Annals of Global Analysis and Geometry **25** (2004) 121–149 \[<a href="https://doi.org/10.1023/B:AGAG.0000018558.65790.db">doi:10.1023/B:AGAG.0000018558.65790.db</a>\] <a href="https://ncatlab.org/nlab/revision/diff/Maslov+index/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Maslov+index/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Maslov+index">current</a>

   added pointer to:

   • Paolo Piccione, Alessandro Portaluri, Daniel V. Tausk: Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics, Annals of Global Analysis and Geometry 25 (2004) 121–149 [doi:10.1023/B:AGAG.0000018558.65790.db]

   diff, v15, current