Journal of the Mathematical Society of Japan

Proof of Kobayashi's rank conjecture on Clifford–Klein forms


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Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space $G/H$ of reductive type does not admit a compact Clifford–Klein form if $\operatorname{rank} G - \operatorname{rank} K < \operatorname{rank} H - \operatorname{rank} K_H$. We solve this conjecture affirmatively. We apply a cohomological obstruction to the existence of compact Clifford–Klein forms proved previously by the author, and use the Sullivan model for a reductive pair due to Cartan–Chevalley–Koszul–Weil.


This work was supported by JSPS KAKENHI Grant Numbers 14J08233 and 17H06784, and the Program for Leading Graduate Schools, MEXT, Japan.

Article information

J. Math. Soc. Japan, Advance publication (2019), 19 pages.

Received: 18 May 2017
Revised: 29 May 2018
First available in Project Euclid: 3 July 2019

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Digital Object Identifier

Primary: 57S30: Discontinuous groups of transformations
Secondary: 17B56: Cohomology of Lie (super)algebras 55P62: Rational homotopy theory 57T15: Homology and cohomology of homogeneous spaces of Lie groups

Clifford–Klein form pure Sullivan algebra relative Lie algebra cohomology transgression


MORITA, Yosuke. Proof of Kobayashi's rank conjecture on Clifford–Klein forms. J. Math. Soc. Japan, advance publication, 3 July 2019. doi:10.2969/jmsj/78007800.

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  • [1] Y. Benoist, Actions propres sur les espaces homogènes réductifs, Ann. of Math. (2), 144 (1996), 315–347.
  • [2] H. Cartan, Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, 15–27; reprinted in [7].
  • [3] H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, 57–71; reprinted in [7].
  • [4] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85–124.
  • [5] Y. Félix, S. Halperin and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, 205, Springer-Verlag, New York, 2001.
  • [6] W. Greub, S. Halperin and R. Vanstone, Connections, curvature, and cohomology, Volume III: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, 47-III, Academic Press, New York–London, 1976.
  • [7] V. W. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999.
  • [8] T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann., 285 (1989), 249–263.
  • [9] T. Kobayashi, Homogeneous spaces with indefinite-metric and discontinuous groups, The 36th Geometry Symposium in Japan, 1989, 104–116 (in Japanese), available at:
  • [10] T. Kobayashi, A necessary condition for the existence of compact Clifford–Klein forms of homogeneous spaces of reductive type, Duke Math. J., 67 (1992), 653–664.
  • [11] T. Kobayashi and K. Ono, Note on Hirzebruch's proportionality principle, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 37 (1990), 71–87.
  • [12] G. Margulis, Existence of compact quotients of homogeneous spaces, measurably proper actions, and decay of matrix coefficients, Bull. Soc. Math. France, 125 (1997), 447–456.
  • [13] Y. Morita, A topological necessary condition for the existence of compact Clifford–Klein forms, J. Differential Geom., 100 (2015), 533–545.
  • [14] Y. Morita, Homogeneous spaces of nonreductive type that do not model any compact manifold, Publ. Res. Inst. Math. Sci., 53 (2017), 287–298.
  • [15] A. L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994.
  • [16] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 47 (1977), 269–331.
  • [17] N. Tholozan, Volume and non-existence of compact Clifford–Klein forms, arXiv:1511.09448v2, preprint.
  • [18] N. Tholozan, Chern–Simons theory and cohomological invariant of character varieties, in preparation.
  • [19] R. J. Zimmer, Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc., 7 (1994), 159–168.