Degrees of maps between locally symmetric spaces

Abstract

Abstract. Let X be a locally symmetric space Γ\G/K where G is a connected noncompact semisimple real Lie group with trivial centre, K is a maximal compact subgroup of G, and Γ ⊂ G is a torsion-free irreducible lattice in G. Let Y = Λ\H/L be another such space having the same dimension as X. Suppose that real rank of G is at least 2. We show that any f : X → Y is either null-homotopic or is homotopic to a covering projection of degree an integer that depends only on Γ and Λ. As a corollary we obtain that the set [X, Y ] of homotopy classes of maps from X to Y is finite.We obtain results on the (non-) existence of orientation reversing diffeomorphisms on X as well as the fixed point property for X.