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 nonpositive 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).
57Q99; 57M99
IntroductionThe complexity c.M / of a compact 3-manifold M (possibly with boundary) was defined in a nice paper of Matveev [29] as the minimum number of vertices of an almost simple spine of M . In that paper he proved the following properties:Finiteness There are finitely many closed irreducible (or cusped hyperbolic) 3-manifolds of bounded complexity.Monotonicity If M S is obtained by cutting M along an incompressible surface S , then c.M S / 6 c.M /.Thanks to the combinatorial nature of spines, it is not hard to classify all manifolds having increasing complexity 0; 1; 2; : : :. Tables have been produced Table 1 below). Some of these classifications were actually done using the dual viewpoint of singular triangulations, which turns out to be equivalent to Matveev's for the most interesting 3-manifolds.We extend here Matveev's complexity from dimension 3 to arbitrary dimension. To do this, we need to choose an appropriate notion of spine. In another paper [27] written in 1973, Matveev defined and studied simple spines of manifolds in arbitrary dimension. A simple spine of a compact manifold is a (locally flat) codimension-1 subpolyhedron with generic singularities, onto which the manifold collapses. If the manifold is closed there cannot be any collapse at all and we therefore need to priorly remove one ball.Simple spines are actually not flexible enough for defining a good complexity. In dimension 3, as an example, any simple spine for S 3 (or, equivalently, D 3 ) is a complicated and unnatural object, such as Bing's house or the abalone. Every simple spine of D 3 has at least one vertex. However, a reasonable complexity must be zero on discs and spheres.To gain more flexibility, Matveev defined in 1988 the more general class of almost simple spines [28; 29] of 3-manifolds. An almost simple polyhedron is a compact polyhedron that can be locally embedded in a simple one. This more general definition allows one to use very natural objects as spines, such as a point for D 3 or a circle for D 2 S 1 : a point is not a vertex by definition, ...