Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

See all formats and pricing
More options …
Volume 2018, Issue 745


Equivariant basic cohomology of Riemannian foliations

Oliver Goertsches / Dirk Töben
  • Departamento de Matemática, Universidade Federal de São Carlos, Rod. Washington Luís, Km 235, C.P. 676, 13565-905 São Carlos, SP, Brazil
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-02-20 | DOI: https://doi.org/10.1515/crelle-2015-0102


The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.


  • [1]

    C. Allday and V. Puppe, Cohomological methods in transformation groups, Cambridge Stud. Adv. Math. 32, Cambridge University Press, Cambridge 1993. Google Scholar

  • [2]

    J. Alvarez López, Morse inequalities for pseudogroups of local isometries, J. Differential Geom. 37 (1993), no. 3, 603–638. CrossrefGoogle Scholar

  • [3]

    J. Alvarez López and Y. Kordykov, Lefschetz distribution of Lie foliations, C*-algebras and elliptic theory II (Bȩdlewo 2006), Trends Math., Birkhäuser, Basel (2008), 1–40. Google Scholar

  • [4]

    M. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. CrossrefGoogle Scholar

  • [5]

    V. Belfi, E. Park and K. Richardson, A Hopf index theorem for foliations, Differential Geom. Appl. 18 (2003), no. 3, 319–341. CrossrefGoogle Scholar

  • [6]

    A. Borel, G. Bredon, E. Floyd, P. Montgomery and R. Palais, Seminar on transformation groups, Ann. of Math. Stud. 46, Princeton University Press, Princeton 1960. Google Scholar

  • [7]

    R. Bott, Vector fields and characteristic numbers, Michigan Math. J. 14 (1967), 231–244. CrossrefGoogle Scholar

  • [8]

    W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge 1993. Google Scholar

  • [9]

    H. Cartan, Cohomologie réelle d’un espace fibré principal differentiable, Sémin. Cartan 2 (1949/1950), Exp. No. 19–20. Google Scholar

  • [10]

    H. Cartan, Notions d’algèbre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de topologie (Bruxelles 1950), CBRM, Liège (1951), 15–27. Google Scholar

  • [11]

    M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451. CrossrefGoogle Scholar

  • [12]

    J. Duflot, Smooth toral actions, Topology 22 (1983), 253–265. CrossrefGoogle Scholar

  • [13]

    D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math. 150, Springer, New York 1995. Google Scholar

  • [14]

    D. Eisenbud, The geometry of syzygies, Grad. Texts in Math. 229, Springer, New York 2004. Google Scholar

  • [15]

    A. El Kacimi-Alaoui, V. Sergiescu and G. Hector, La cohomologie basique d’un feuilletage riemannien est de dimension finie, Math. Z. 188 (1985), no. 4, 593–599. CrossrefGoogle Scholar

  • [16]

    M. Franz and V. Puppe, Exact sequences for equivariantly formal spaces, C. R. Math. Acad. Sci. Soc. R. Can. 33 (2011), no. 1, 1–10. Google Scholar

  • [17]

    É. Ghys, Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 4, 203–223. CrossrefGoogle Scholar

  • [18]

    O. Goertsches and D. Töben, Torus actions whose equivariant cohomology is Cohen–Macaulay, J. Topol. 3 (2010), no. 4, 819–846. CrossrefWeb of ScienceGoogle Scholar

  • [19]

    O. Goertsches, H. Nozawa and D. Töben, Equivariant cohomology of K-contact manifolds, Math. Ann. 354 (2012), no. 4, 1555–1582. CrossrefWeb of ScienceGoogle Scholar

  • [20]

    M. Goresky, R. Kottwitz and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. Google Scholar

  • [21]

    V. Guillemin, V. Ginzburg and Y. Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Math. Surveys Monogr. 96, American Mathematical Society, Providence 2002. Google Scholar

  • [22]

    V. Guillemin, E. Lerman and S. Sternberg, Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge 1996. Google Scholar

  • [23]

    V. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer, Berlin 1999. Google Scholar

  • [24]

    W.-Y. Hsiang, Cohomology theory of topological transformation groups, Ergeb. Math. Grenzgeb. 85, Springer, Berlin 1975. Google Scholar

  • [25]

    S. Hurder and D. Töben, Transverse LS category for Riemannian foliations, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5647–5680. CrossrefGoogle Scholar

  • [26]

    F. Kamber and P. Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Math. 493, Springer, Berlin 1975. Google Scholar

  • [27]

    F. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes 31, Princeton University Press, Princeton 1984. Google Scholar

  • [28]

    J. McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Stud. Adv. Math. 58, Cambridge University Press, Cambridge 2001. Google Scholar

  • [29]

    P. Molino, Riemannian foliations. With appendices by G. Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu, Birkhäuser, Boston 1988. Google Scholar

  • [30]

    W. Mozgawa, Feuilletages de Killing, Collect. Math. 36 (1985), no. 3, 285–290. Google Scholar

  • [31]

    B. Reinhart, Harmonic integrals on foliated manifolds, Amer. J. Math. 81 (1959), 529–536. CrossrefGoogle Scholar

  • [32]

    É. Salem, Une généralisation du théorème de Myers–Steenrod aux pseudogroupes d’isométries, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 2, 185–200. CrossrefGoogle Scholar

  • [33]

    I. Satake, On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. USA 42 (1956), 359–363. CrossrefGoogle Scholar

  • [34]

    J.-P. Serre, Local algebra, Springer Monogr. Math., Springer, Berlin 2000. Google Scholar

  • [35]

    P. Stefan, Accessibility and foliations with singularities, Bull. Amer. Math. Soc. 80 (1974), 1142–1145. CrossrefGoogle Scholar

  • [36]

    H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188. CrossrefGoogle Scholar

  • [37]

    S. Tolman and J. Weitsman, On the cohomology rings of Hamiltonian T-spaces, Northern California symplectic geometry seminar, Amer. Math. Soc. Transl. Ser. 2 196, American Mathematical Society, Providence (1999), 251–258. Google Scholar

About the article

Received: 2012-06-22

Revised: 2015-10-08

Published Online: 2016-02-20

Published in Print: 2018-12-01

Dirk Töben was supported by the Schwerpunktprogramm SPP 1154 of the DFG.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 745, Pages 1–40, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0102.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Oliver Goertsches and Leopold Zoller
São Paulo Journal of Mathematical Sciences, 2019
Yi Lin and Xiangdong Yang
Pacific Journal of Mathematics, 2019, Volume 298, Number 1, Page 59
Oliver Goertsches, Hiraku Nozawa, and Dirk Töben
Israel Journal of Mathematics, 2017, Volume 222, Number 2, Page 867

Comments (0)

Please log in or register to comment.
Log in