Degrees of maps between locally symmetric spaces

Abstract

Let X be a locally symmetric space Γ\\G/K where G is a connected non-compact 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 it 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.