Modification and the cohomology groups of compact solvmanifolds II

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Abstract

In this article, we refine the modification theorem for a compact solvmanifold given in 2006 and completely solve the problem of finding the cohomology ring on compact solvmanifolds.

Keywords

solvmanifolds / cohomology / invariant structure / homogeneous space / product / fiber bundles / symplectic manifolds / splittings / prealgebraic group / decompositions / modification / Lie group / compact manifolds / uniform discrete subgroups / locally flat parallelizable manifolds

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A Correction to this paper has been published:
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This article has been retracted.Please see the Retraction Notice for more detail:
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References

[1] G. D. Mostow, Cohomology of topological groups and solvmanifolds, Ann. Math., 73 (1961), 20-48. https://doi.org/10.2307/1970281

[2] M. S. Raghunathan, Discrete Subgroups of Lie Groups , Springer, Belin, 1972.

[3] D. Guan, Modification and the cohomology groups of compact solvmanifolds, Electron. Res. Announce. Am. Math. Soc., 13 (2007), 74-81. http://doi.org/10.1090/S1079-6762-07-00176-X

[4] D. Guan, Classification of compact complex homogeneous manifolds with pseudo-kählerian structures, J. Algebra, 324 (2010), 2010-2024. https://doi.org/10.1016/j.jalgebra.2010.06.013

[5] A. L. Onishchik, E. B. Vinberg, Lie Groups and Lie algebras III: Structure of Lie Groups and Lie Algebras , Springer, Berlin, 1994.

[6] A. Hattori, Spectral sequences in the de rham cohomology of the fiber bundles, J. Fac. Sci. Univ. Tokyo, 8 (1960), 289-331.

[7] T. Yamada, A pseudo-kähler structure on a nontoral compact complex parallelizable solvmanifold, Geom. Dedicata, 112 (2005), 115-122. https://doi.org/10.1007/s10711-004-3397-4

[8] D. Guan, Toward a classification of real compact solvmanifolds with real symplectic structures, J. Algebra, 379 (2013), 144-155. https://doi.org/10.1016/j.jalgebra.2013.01.011

[9] C. Benson, C. S. Gordon, Kähler structures on compact solvmanifolds, Proc. Am. Math. Soc. 108 (1990), 971-980.

[10] S. Console, A. Fino, On the de Rham cohomology of solvmanifolds, Ann. Sc. Norm. Super. Pisa-Cl. Sci., 10 (2011), 801-818.

[11] D. Witte, Superrigidity of lattices in solvable Lie groups, Invent. Math., 122 (1995), 147-193. https://doi.org/10.1007/BF01231442

[12] D. Witte, Archimedean superrigidity of solvable S -Arithmetic groups , J. Algebra, 187 (1997), 268-288. https://doi.org/10.1006/jabr.1997.6785

[13] H. Kasuya, Cohomologically symplectic solvmanifolds are symplectic, J. Symplect. Geom., 9 (2011), 429-434. http://doi.org/10.4310/JSG.2011.v9.n4.a1

[14] H. Kasuya, Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems, J. Differ. Geom., 93 (2013), 267-297. https://doi.org/10.4310/jdg/1361800867

[15] N. Dungey, A. F. M. Elst, D. W. Robinson, Analysis on Lie Groups with Polynomial Growth , Birkhäuser, Boston, 2003. https://doi.org/10.1007/978-1-4612-2062-6

[16] R. Azencott, E. N. Wilso, Homogeneous Manifolds with Negative Curvatures, Part II , American Mathematical Society, Rhode Island, 1976.

[17] V. V. Gorbatservich, Plesiocompact homogeneous spaces, Sib. Math. J., 30 (1989), 217-226. https://doi.org/10.1007/BF00971376

[18] J. Dorfmeister, Homogeneous kähler manifolds admitting a transitive solvable group of automorphisms, Ann. Sci. Ec. Norm. Super., 18 (1985), 143-180. https://doi.org/10.24033/asens.1487

[19] J. Dorfmeister, Quasi Clans, Abh. Math. Semin. Univ. Hambg., 50 (1980), 178-187. https://doi.org/10.1007/BF02941427