In his paper, 'On torsion in the cohomology of locally symmetric varieties', Peter Scholze has introduced (for unitary or symplectic groups) a purely topological construction of the 'Eisenstein' cohomology classes on arithmetic quotients of the symmetric spaces of a reductive group over Q, associated to the inner cohomology of the arithmetic quotients of the symmetric spaces of its maximal Levi subgroups. This relies purely on the Borel-Serre compactification. I will explain the extension of this construction to the general case, and to local systems. This seems to yield such a construction in cases which have not been obtained by the theory of Eisenstein cohomology; I will explain the relation in the simple case of GL(n). The construction works essentially with arbitrary coefficients; in particular it gives a direct construction of classes over Q or its algebraic closure, restricting as expected to suitable cells of the Borel-Serre compactification.

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