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nLab > Latest Changes: Maslov index

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeNov 10th 2013
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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
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>LGrass</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^1(LGrass, \mathbb{Z}) \simeq \mathbb{Z} \,. </annotation></semantics></math>$
The generator of this cohomology group is called the universal Maslov index
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>u</mi><mo>∈</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>LGrass</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> u \in H^1(LGrass, \mathbb{Z}) \,. </annotation></semantics></math>$
Given a Lagrangian submanifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \hookrightarrow X</annotation></semantics></math>$ of a symplectic manifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math>$ , its tangent bundle is classified by a function
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>i</mi><mspace width="0.27778em"/><mo lspace="0.11111em">:</mo><mspace width="0.27778em"/><mi>Y</mi><mo>→</mo><mi>LGrass</mi><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> i \;\colon\; Y \to LGrass \,. </annotation></semantics></math>$
The _Maslov index of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>$ is the universal Maslov index pulled back along this map
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>u</mi><mo>∈</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="0.16667em"/><mo>.</mo></mrow><annotation encoding="application/x-tex"> i^\ast u \in H^1(Y,\mathbb{Z}) \,. </annotation></semantics></math>$
- 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
- 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
- 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

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nLab > Latest Changes: Maslov index