Cohomology of compact locally symmetric spaces

Abstract

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S(Γ)=Γ\X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup Γ of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S_H(Γ)=Δ\Y of Γ\X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in Cⁿ and C^m , then low degree cohomology classes on the variety S(Γ) restrict non-trivially to the subvariety S_H(Γ); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S(Γ).