The main aim of this volume of the ‘Mémoires de la SMF’ is to study properties of cohomology groups for an arithmetic manifold and its totally geodesic submanifolds. The classical Lefschetz hyperplane theorem states that the restriction map induced on suitable cohomology groups by a (generic) hyperplane section of a projective manifold is injective. In the book, the author discusses a (broad) analogy of the classical Lefschetz theorem for arithmetic manifolds associated to the unitary and orthogonal groups U(p,q) and O(p,q). There are two types of results: injectivity of the map associated with a (virtual) restriction and the one associated with ‘a cup product with the class of the cycle’. The methods used are based on representation theory and a review of required results can be found in the early sections of the book.

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