  We consider locally symmetric manifolds with a fixed universal covering, and
construct for each such manifold M a simplicial complex R whose size is
proportional to the volume of M. When M is non-compact, R is homotopically
equivalent to M, while when M is compact, R is homotopically equivalent to M\N,
where N is a finite union of submanifolds of fairly smaller dimensions. This
expresses how the volume controls the topological structure of M, and yields
concrete bounds for various finiteness statements which previously had no
quantitative proofs. For example, it gives an explicit upper bound for the
possible number of locally symmetric manifolds of volume bounded by v>0, and it
yields an estimate for the size of a minimal presentation for the fundamental
group of a manifold in terms of its volume. It also yields a number of new
finiteness results.
