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Lefschetz properties in the cohomology of certain arithmetic manifolds: the case of Hilbert modular surfaces. (Propriétés de Lefschetz dans la cohomologie de certaines variétés arithmétiques: Le cas des surfaces modulaires de Hilbert.) (French) Zbl 1053.11047
Seminar on spectral theory and geometry. 2002–2003. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Sémin. Théor. Spectr. Géom. 21, 75-101 (2003).
In the papers [N. Bergeron, Lefschetz properties for arithmetic real and complex hyperbolic manifolds. Int. Math. Res. Not. 2003, No. 20, 1089–1122 (2003; Zbl 1036.11021) and N. Bergeron and L. Clozel, Spectre et homologie des variétés hyperboliques complexes de congruence. C. R., Math., Acad. Sci. Paris 334, 995–998 (2002; Zbl 1011.32020)] the authors stated several conjectures connecting the (co-)homology of a hyperbolic manifold with the (co-) homology of a totally geodesic submanifold. In these papers partial results are also proved.
In the current paper, the author illustrates the techniques of [the above cited papers] in the simple case of a Hilbert modular surface $$M$$. More precisely, he shows that for a given totally geodesic submanifold $$S\subset M$$ there exists a finite covering $$\widetilde M$$ of $$M$$ such that the embedding of $$S$$ lifts to $$M$$ and the class $$[S]$$ is non-zero in $$H_2(\widetilde M)$$.
For the entire collection see [Zbl 1032.35005].
MSC:
 11F75 Cohomology of arithmetic groups 53C35 Differential geometry of symmetric spaces 57T15 Homology and cohomology of homogeneous spaces of Lie groups
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