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Morita, Yosuke

Proof of Kobayashi’s rank conjecture on Clifford-Klein forms. (English) Zbl 1479.57079

J. Math. Soc. Japan 71, No. 4, 1153-1171 (2019).
Suppose \(G/H\) is a homogeneous space of reductive type and let \(K\subseteq G\) and \(K_H\subseteq H\) be maximal compact subgroups. The author proves that if \(\operatorname{rank}(G) - \operatorname{rank}(K) < \operatorname{rank}(H) - \operatorname{rank}(K_H)\), then \(G/H\) does not admit a compact Clifford-Klein form. That is, there is no discrete subgroup \(\Gamma\subseteq G\) which acts property and freely for which the quotient space \(\Gamma \backslash G/H\) is compact. This affirmatively answers a conjecture of Kobayashi.
The author had previously established [J. Differ. Geom. 100, No. 3, 533–545 (2015; Zbl 1323.53056); Publ. Res. Inst. Math. Sci. 53, No. 2, 287–298 (2017; Zbl 1371.53051)] that non-existence of a compact Clifford-Klein form would follow if a certain map on the level of Lie algebra cohomology was not injective.
The main result of the article under review is that, under the above rank inequalities, this Lie algebra cohomology map is not injective. The author proves this by constructing a spectral sequence and using it to show that the Lie algebra cohomomology map is injective if and only if another map is surjective. However, the rank condition easily implies this latter map is not surjective.
Reviewer: Jason DeVito (Martin)

Cited in 1 Document

MSC:

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

Keywords:

Clifford-Klein form; pure Sullivan algebra; relative Lie algebra cohomology; transgression

Citations:

Zbl 1323.53056; Zbl 1371.53051
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\textit{Y. Morita}, J. Math. Soc. Japan 71, No. 4, 1153--1171 (2019; Zbl 1479.57079)
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References:

[1] Y. Benoist, Actions propres sur les espaces homogènes réductifs, Ann. of Math. (2), 144 (1996), 315-347. · Zbl 0868.22013
[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]. · Zbl 0045.30601
[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. · Zbl 0031.24803
[5] Y. Félix, S. Halperin and J.-C. Thomas, Rational homotopy theory, Graduate Texts in Mathematics, 205, Springer-Verlag, New York, 2001. · Zbl 0961.55002
[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. · Zbl 0372.57001
[7] V. W. Guillemin and S. Sternberg, Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. · Zbl 0934.55007
[8] T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann., 285 (1989), 249-263. · Zbl 0662.22008
[9] T. Kobayashi, Homogeneous spaces with indefinite-metric and discontinuous groups, The 36th Geometry Symposium in Japan, 1989, 104-116 (in Japanese), available at: http://www.ms.u-tokyo.ac.jp/ toshi/texpdf/tk12-kika-sympo89.pdf.
[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. · Zbl 0799.53056
[11] T. Kobayashi and K. Ono, Note on Hirzebruch’s proportionality principle, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 37 (1990), 71-87. · Zbl 0726.57019
[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. · Zbl 0892.22009
[13] Y. Morita, A topological necessary condition for the existence of compact Clifford-Klein forms, J. Differential Geom., 100 (2015), 533-545. · Zbl 1323.53056
[14] Y. Morita, Homogeneous spaces of nonreductive type that do not model any compact manifold, Publ. Res. Inst. Math. Sci., 53 (2017), 287-298. · Zbl 1371.53051
[15] A. L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994. · Zbl 0796.57001
[16] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math., 47 (1977), 269-331. · Zbl 0374.57002
[17] N. Tholozan, Volume and non-existence of compact Clifford-Klein forms, arXiv:1511.09448v2, preprint. · Zbl 1375.53022
[18] N. Tholozan, Chern-Simons theory and cohomological invariant of character varieties, in preparation. · Zbl 1375.53022
[19] R. J. Zimmer, Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc., 7 (1994), 159-168. · Zbl 0801.22009
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