The Godbillon-Vey invariant of a transversely homogeneous foliation
HTML articles powered by AMS MathViewer
- by Robert Brooks and William Goldman PDF
- Trans. Amer. Math. Soc. 286 (1984), 651-664 Request permission
Abstract:
A real projective foliation is a foliation $\mathfrak {F}$ with a system of local coordinates transverse to $\mathfrak {F}$ modelled on ${\mathbf {R}}{P^1}$ (so that the coordinate changes are real linear fractional transformations). Given a closed manifold $M$, there is but a finite set of values in ${H^3}(M;{\mathbf {R}})$ which the Godbillon-Vey invariant of such foliations may assume. A bound on the possible values, in terms of the fundamental group, is computed. For $M$ an oriented circle bundle over a surface, this finite set is explicitly computed.References
- Robert A. Blumenthal, Transversely homogeneous foliations, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, vii, 143–158 (English, with French summary). MR 558593 R. Brooks, On the smooth cohomology of groups of diffeomorphisms, Ph.D. thesis, Harvard University, 1977.
- Robert Brooks, Volumes and characteristic classes of foliations, Topology 18 (1979), no. 4, 295–304. MR 551011, DOI 10.1016/0040-9383(79)90020-X
- Johan L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), no. 3, 233–245. MR 413122, DOI 10.1016/0040-9383(76)90038-0
- Claude Godbillon and Jacques Vey, Un invariant des feuilletages de codimension $1$, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A92-A95 (French). MR 283816 W. Goldman, Discontinuous groups and the Euler class, Ph.D. thesis, University of California, Berkeley, 1980.
- Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
- James L. Heitsch, Secondary invariants of transversely homogeneous foliations, Michigan Math. J. 33 (1986), no. 1, 47–54. MR 817908, DOI 10.1307/mmj/1029003289
- John Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215–223. MR 95518, DOI 10.1007/BF02564579
- Dennis Sullivan, A generalization of Milnor’s inequality concerning affine foliations and affine manifolds, Comment. Math. Helv. 51 (1976), no. 2, 183–189. MR 418119, DOI 10.1007/BF02568150
- William Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304–307. MR 339267, DOI 10.1090/S0002-9904-1974-13475-0 —, Non-cobordant foliations of ${S^3}$, Bull. Amer. Math. Soc. 78 (1972), 511-514. —, Foliations of three-manifolds which are circle bundles, Ph.D. thesis, University of California, Berkeley, 1972.
- W. T. van Est, Une application d’une méthode de Cartan-Leray, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 542–544 (French). MR 0073108
- John W. Wood, Foliated $S^{1}$-bundles and diffeomorphisms of $S^{1}$, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 671–681. MR 0343294
- H. Blaine Lawson Jr., Foliations, Bull. Amer. Math. Soc. 80 (1974), 369–418. MR 343289, DOI 10.1090/S0002-9904-1974-13432-4
- Raoul Bott, On some formulas for the characteristic classes of group-actions, Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontifícia Univ. Católica, Rio de Janeiro, 1976) Lecture Notes in Mathematics, Vol. 652, Springer, Berlin, 1978, pp. 25–61. MR 505649
- Morris W. Hirsch and William P. Thurston, Foliated bundles, invariant measures and flat manifolds, Ann. of Math. (2) 101 (1975), 369–390. MR 370615, DOI 10.2307/1970996
- Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR 24908, DOI 10.1090/S0002-9947-1948-0024908-8
- Robert Brooks and William Goldman, Volumes in Seifert space, Duke Math. J. 51 (1984), no. 3, 529–545. MR 757951, DOI 10.1215/S0012-7094-84-05126-3
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 286 (1984), 651-664
- MSC: Primary 53C12; Secondary 55R40, 57R32
- DOI: https://doi.org/10.1090/S0002-9947-1984-0760978-9
- MathSciNet review: 760978