  We extend Matveev's complexity of 3-manifolds to PL compact manifolds of
arbitrary dimension, and we study its properties. The complexity of a manifold
is the minimum number of vertices in a simple spine. We study how this quantity
changes under the most common topological operations (handle additions, finite
coverings, drilling and surgery of spheres, products, connected sums) and its
relations with some geometric invariants (Gromov norm, spherical volume, volume
entropy, systolic constant).
  Complexity distinguishes some homotopically equivalent manifolds and is
positive on all closed aspherical manifolds (in particular, on manifolds with
non-positive sectional curvature). There are finitely many closed hyperbolic
manifolds of any given complexity. On the other hand, there are many closed
4-manifolds of complexity zero (manifolds without 3-handles, doubles of
2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with
arbitrary fundamental group).
