Residue classes of Lagrangian subbundles and Maslov classes

Suzuki, Haruo

doi:10.1090/S0002-9947-1995-1282897-1

Residue classes of Lagrangian subbundles and Maslov classes
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Trans. Amer. Math. Soc. 347 (1995), 189-202 Request permission

Abstract:

For Lagrangian subbundles with singularities in symplectic vector bundles, explicit formulas of relation between their residue classes and Maslov classes outside singularities are obtained. Then a Lagrangian subbundle with singularity is constructed where all possible Maslov classes are nonzero but residue classes vanish for dimension $> 2$. Moreover, a Lagrangian immersion with singularity is constructed, where the similar property for the associated Maslov classes and residue classes is shown.

References

Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304, DOI 10.1007/978-1-4757-3951-0
Shiing Shen Chern and James Simons, Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48–69. MR 353327, DOI 10.2307/1971013
Franz W. Kamber and Philippe Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin-New York, 1975. MR 0402773, DOI 10.1007/BFb0081558
Daniel Lehmann, Variétés stratifiées $C^\infty$: intégration de Čech-de Rham, et théorie de Chern-Weil, Geometry and topology of submanifolds, II (Avignon, 1988) World Sci. Publ., Teaneck, NJ, 1990, pp. 205–248 (French). MR 1068742
D. Lehmann, Classes caractéristiques residuelles, Differential geometry and its applications (Brno, 1989) World Sci. Publ., Teaneck, NJ, 1990, pp. 85–108 (French, with English summary). MR 1062009
John McCleary, User’s guide to spectral sequences, Mathematics Lecture Series, vol. 12, Publish or Perish, Inc., Wilmington, DE, 1985. MR 820463
J.-M. Morvan and L. Niglio, Classes caractéristiques des couples de sous-fibrés lagrangiens, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 2, 193–209 (French, with English summary). MR 850751, DOI 10.5802/aif.1055
Jean-Pierre Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425–505 (French). MR 45386, DOI 10.2307/1969485
Haruo Suzuki, Chern-Simons-Maslov classes of some symplectic vector bundles, Proc. Amer. Math. Soc. 117 (1993), no. 2, 541–546. MR 1124152, DOI 10.1090/S0002-9939-1993-1124152-X
Izu Vaisman, Symplectic geometry and secondary characteristic classes, Progress in Mathematics, vol. 72, Birkhäuser Boston, Inc., Boston, MA, 1987. MR 932470, DOI 10.1007/978-1-4757-1960-4
André Weil, Sur les théorèmes de de Rham, Comment. Math. Helv. 26 (1952), 119–145 (French). MR 50280, DOI 10.1007/BF02564296