Cohomology of locally symmetric spaces[HBNI Th116]

Abstract

Our object of study has been the topology of locally symmetric spaces of non-compact type. The thesis consists of two parts. The first part addresses homotopy classification of maps between higher rank irreducible locally symmetric spaces including possible degrees in terms of the lattices involved. In particular we have addressed the question of when the degree can be negative. The second part involves construction of cohomology classes of a family of compact locally symmetric spaces associated to SO ∗ (2n). These classes are Poincare duals of certain totally geodesic submanifolds. Using this, we detect occurrence of certain irreducible unitary representations associated to θ-stable parabolic subalgebras of g, in the direct Hilbert sum decomposition of L 2 (Γ\SO ∗ (2n)).