Reduction of symplectic homeomorphisms
[Réduction des homéomorphismes symplectiques]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 3, pp. 633-668.

Nous avons démontré dans [9], qu'un homéomorphisme symplectique qui laisse invariante une sous-variété coïsotrope C, préserve également son feuilletage caractéristique. Il induit donc un homéomorphisme sur la réduction symplectique de C.

Dans cet article, nous démontrons que l'homéomorphisme ainsi obtenu exhibe certaines propriétés symplectiques. En particulier, dans le cas où la variété symplectique ambiante est un tore et la sous-variété coïsotrope est un sous-tore standard, nous démontrons que l'homéomorphisme réduit préserve les invariants spectraux et donc aussi la capacité spectrale.

Pour démontrer notre résultat principal, nous construisons, à l'aide de l'homologie de Floer lagrangienne, une nouvelle famille d'invariants spectraux qui satisfont un nouveau type d'inégalité triangulaire.

In [9], we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C.

In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity.

To prove our main result, we use Lagrangian Floer theory to construct a new class of spectral invariants which satisfy a non-standard triangle inequality.

Publié le :
DOI : 10.24033/asens.2292
Classification : 53D40; 37J05.
Keywords: Symplectic manifolds, symplectic reduction, $C^0$--symplectic topology, spectral invariants.
Mot clés : Variétés symplectiques, réduction symplectique, topologie symplectique $C^0$, invariants spectraux. 
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     title = {Reduction of symplectic homeomorphisms},
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     pages = {633--668},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Humilière, Vincent; Leclercq, Rémi; Seyfaddini, Sobhan. Reduction of symplectic homeomorphisms. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 3, pp. 633-668. doi : 10.24033/asens.2292. http://archive.numdam.org/articles/10.24033/asens.2292/

Albers, P. A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not., Volume 2008 (2008) (ISSN: 1073-7928) | DOI | MR | Zbl

Abbondandolo, A.; Schwarz, M. Floer homology of cotangent bundles and the loop product, Geom. Topol., Volume 14 (2010), pp. 1569-1722 (ISSN: 1465-3060) | DOI | MR | Zbl

Auroux, D. A beginner's introduction to Fukaya categories, Contact and symplectic topology (Bourgeois, F. et al., eds.) (Bolyai Soc. Math. Stud.), Volume 26 (2014), pp. 85-136 | DOI | MR | Zbl

Barraud, J.-F.; Cornea, O., Morse theoretic methods in nonlinear analysis and in symplectic topology (NATO Sci. Ser. II Math. Phys. Chem.), Volume 217, Springer, Dordrecht, 2006, pp. 109-148 | DOI | MR | Zbl

Buhovsky, L.; Opshtein, E. Some quantitative results in C 0 symplectic geometry (preprint arXiv:1404.0875, to appear in Invent. math ) | MR

Entov, M.; Polterovich, L. Rigid subsets of symplectic manifolds, Compos. Math., Volume 145 (2009), pp. 773-826 (ISSN: 0010-437X) | DOI | MR | Zbl

Hu, S.; Lalonde, F.; Leclercq, R. Homological Lagrangian monodromy, Geom. Topol., Volume 15 (2011), pp. 1617-1650 (ISSN: 1465-3060) | DOI | MR | Zbl

Humilière, V.; Leclercq, R.; Seyfaddini, S. Coisotropic rigidity and C0-symplectic geometry, Duke Math. J., Volume 164 (2015), pp. 767-799 (ISSN: 0012-7094) | DOI | MR | Zbl

Humilière, V. On some completions of the space of Hamiltonian maps, Bull. Soc. Math. France, Volume 136 (2008), pp. 373-404 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Katić, J.; Milinković, D. Piunikhin-Salamon-Schwarz isomorphisms for Lagrangian intersections, Differential Geom. Appl., Volume 22 (2005), pp. 215-227 (ISSN: 0926-2245) | DOI | MR | Zbl

Leclercq, R. Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn., Volume 2 (2008), pp. 249-286 (ISSN: 1930-5311) | DOI | MR | Zbl

Leclercq, R. The Seidel morphism of Cartesian products, Algebr. Geom. Topol., Volume 9 (2009), pp. 1951-1969 (ISSN: 1472-2747) | DOI | MR | Zbl

Laudenbach, F.; Sikorav, J.-C. Hamiltonian disjunction and limits of Lagrangian submanifolds, Int. Math. Res. Not., Volume 1994 (1994), pp. 161-168 (ISSN: 1073-7928) | DOI | MR | Zbl

Monzner, A.; Vichery, N.; Zapolsky, F. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization, J. Mod. Dyn., Volume 6 (2012), pp. 205-249 | MR | Zbl

Oh, Y.-G. Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Differential Geom., Volume 46 (1997), pp. 499-577 http://projecteuclid.org/getRecord?id=euclid.jdg/1214459976 (ISSN: 0022-040X) | MR | Zbl

Oh, Y.-G.; Müller, S. The group of Hamiltonian homeomorphisms and C0–symplectic topology, J. Symplectic Geom., Volume 5 (2007), pp. 167-219 http://projecteuclid.org/getRecord?id=euclid.jsg/1202004455 (ISSN: 1527-5256) | DOI | MR | Zbl

Opshtein, E. C0–rigidity of characteristics in symplectic geometry, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009), pp. 857-864 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Piunikhin, S.; Salamon, D.; Schwarz, M., Contact and symplectic geometry (Cambridge, 1994) (Publ. Newton Inst.), Volume 8, Cambridge Univ. Press, 1996, pp. 171-200 | MR | Zbl

Schwarz, M. On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., Volume 193 (2000), pp. 419-461 (ISSN: 0030-8730) | DOI | MR | Zbl

Sabloff, J. M.; Traynor, L. Obstructions to the existence and squeezing of Lagrangian cobordisms, J. Topol. Anal., Volume 2 (2010), pp. 203-232 (ISSN: 1793-5253) | DOI | MR | Zbl

Théret, D. A Lagrangian camel, Comment. Math. Helv., Volume 74 (1999), pp. 591-614 (ISSN: 0010-2571) | DOI | MR | Zbl

Viterbo, C. Symplectic topology as the geometry of generating functions, Math. Annalen, Volume 292 (1992), pp. 685-710 | DOI | MR | Zbl

Zapolsky, F. On the Hofer geometry for weakly exact Lagrangian submanifolds, J. Symplectic Geom., Volume 11 (2013), pp. 475-488 http://projecteuclid.org/euclid.jsg/1384282845 (ISSN: 1527-5256) | DOI | MR | Zbl

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