  We formulate a conjecture that arithmetic locally symmetric manifolds have
simple homotopy type, and prove it for the non-compact case. More precisely, we
show that, for any symmetric space S of non-compact type without Euclidean de
Rham factors, there are constants a=a(S) and d=d(S) such that any non-compact
arithmetic manifold, locally isometric to S, is homotopically equivalent to a
simplicial complex whose vertices degrees are bounded by d, and its number of
vertices is bounded by a times the Riemannian volume. It is very likely that
such a result holds also for compact arithmetic manifolds.
  We conclude that, for any fixed universal covering, S, other then the
hyperbolic plane, there are at most V^(CV) irreducible non-compact arithmetic
manifolds with volume <=V, where C=C(S) is a constant depending on S. Since
higher rank irreducible locally symmetric manifolds of finite volume are always
arithmetic, our result quantifies the number of them which are non-compact.
