Localization of basic characteristic classes

Dirk Töben[1]

  • [1] Universidade Federal de São Carlos Departamento de Matemàtica Rod. Washington Luís, Km 235, C.P. 676 13565-905 São Carlos, SP (Brazil)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 2, page 537-570
  • ISSN: 0373-0956

Abstract

top
We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold M is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type localization theorem for equivariant basic cohomology.

How to cite

top

Töben, Dirk. "Localization of basic characteristic classes." Annales de l’institut Fourier 64.2 (2014): 537-570. <http://eudml.org/doc/275568>.

@article{Töben2014,
abstract = {We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold $M$ is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type localization theorem for equivariant basic cohomology.},
affiliation = {Universidade Federal de São Carlos Departamento de Matemàtica Rod. Washington Luís, Km 235, C.P. 676 13565-905 São Carlos, SP (Brazil)},
author = {Töben, Dirk},
journal = {Annales de l’institut Fourier},
keywords = {Riemannian foliations; basic cohomology; equivariant cohomology; characteristic classes; localization; Killing foliations; transversely oriented foliations; invariants for foliations; characteristic forms; basic forms; Euler-Pontryagin ring; basic characteristic number; Thom isomorphism; foliated bundle},
language = {eng},
number = {2},
pages = {537-570},
publisher = {Association des Annales de l’institut Fourier},
title = {Localization of basic characteristic classes},
url = {http://eudml.org/doc/275568},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Töben, Dirk
TI - Localization of basic characteristic classes
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 2
SP - 537
EP - 570
AB - We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold $M$ is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type localization theorem for equivariant basic cohomology.
LA - eng
KW - Riemannian foliations; basic cohomology; equivariant cohomology; characteristic classes; localization; Killing foliations; transversely oriented foliations; invariants for foliations; characteristic forms; basic forms; Euler-Pontryagin ring; basic characteristic number; Thom isomorphism; foliated bundle
UR - http://eudml.org/doc/275568
ER -

References

top
  1. A. Alekseev, E. Meinrenken, Equivariant cohomology and the Maurer-Cartan equation, Duke Math. J. 130 (2005), 479-521 Zbl1085.57022MR2184568
  2. Marcos M. Alexandrino, Leonardo Biliotti, Renato H. L. Pedrosa, Lectures on isometric actions, (2008), Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro Zbl1151.22001MR2459324
  3. M. F. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1-28 Zbl0521.58025MR721448
  4. Victor Belfi, Efton Park, Ken Richardson, A Hopf index theorem for foliations, Differential Geom. Appl. 18 (2003), 319-341 Zbl1041.53030MR1975032
  5. Nicole Berline, Michèle Vergne, Zéros d’un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J. 50 (1983), 539-549 Zbl0515.58007MR705039
  6. Raoul Bott, Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231-244 Zbl0145.43801MR211416
  7. Raoul Bott, Loring W. Tu, Differential forms in algebraic topology, 82 (1982), Springer-Verlag, New York-Berlin Zbl0496.55001MR658304
  8. Manfredo Perdigão do Carmo, Riemannian geometry, (1992), Birkhäuser Boston, Inc., Boston, MA Zbl0752.53001MR1138207
  9. A. El Kacimi-Alaoui, M. Nicolau, On the topological invariance of the basic cohomology, Math. Ann. 295 (1993), 627-634 Zbl0793.57016MR1214951
  10. O. Goertsches, D. Töben, Equivariant basic cohomolog of Riemannian foliations Zbl1208.55005
  11. Werner Greub, Stephen Halperin, Ray Vanstone, Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles, (1972), Academic Press, New York-London Zbl0322.58001MR336650
  12. Victor W. Guillemin, Shlomo Sternberg, Supersymmetry and equivariant de Rham theory, (1999), Springer-Verlag, Berlin Zbl0934.55007MR1689252
  13. Steven Hurder, Dirk Töben, The equivariant LS-category of polar actions, Topology Appl. 156 (2009), 500-514 Zbl1186.55001MR2492297
  14. Steven Hurder, Dirk Töben, Transverse LS category for Riemannian foliations, Trans. Amer. Math. Soc. 361 (2009), 5647-5680 Zbl1200.57021MR2529908
  15. Franz W. Kamber, Philippe Tondeur, Foliated bundles and characteristic classes, (1975), Springer-Verlag, Berlin-New York Zbl0308.57011MR402773
  16. Connor Lazarov, Joel Pasternack, Residues and characteristic classes for Riemannian foliations, J. Differential Geometry 11 (1976), 599-612 Zbl0367.57005MR445514
  17. John McCleary, A user’s guide to spectral sequences, 58 (2001), Cambridge University Press, Cambridge Zbl0959.55001MR1793722
  18. Pierre Molino, Riemannian foliations, 73 (1988), Birkhäuser Boston, Inc., Boston, MA Zbl0633.53001MR932463
  19. Witold Mozgawa, Feuilletages de Killing, Collect. Math. 36 (1985), 285-290 Zbl0614.53030MR868544
  20. Bruce L. Reinhart, Harmonic integrals on foliated manifolds, Amer. J. Math. 81 (1959), 529-536 Zbl0088.07902MR107280
  21. Takashi Sakai, Riemannian geometry, 149 (1996), American Mathematical Society, Providence, RI Zbl0886.53002MR1390760
  22. Vlad Sergiescu, Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier (Grenoble) 35 (1985), 137-158 Zbl0563.57012MR810671

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.