Complexity of 4-Manifolds

A natural notion of complexity of a PL n-dimensional manifold is the minimal number of highest-dimensional simplices in a triangulation of the manifold. Such a complexity is an integer-valued function and is finite (for each k ≥ 0 there are only finitely many manifolds whose complexity is less than or equal to k). In order to find all the n-manifolds of complexity k, one has to identify all the possible ways of gluing k copies of the n-simplex such that the link of each point is an (n−1)-sphere. Hence, producing lists of low-complexity n-manifolds can be a difficult task if n ≥ 3 because of the sphere-recognition problem. In dimension 3, S. Matveev [Matveev 90] defined an alternative notion of complexity that for “most” 3-manifolds is equivalent to the one defined above. Matveev’s complexity is based on a combinatorial description of 3manifolds by means of 2-polyhedra (their “spines”), and it turns out to be strictly related to the topological properties of the manifolds: for instance, it is additive under connected sums and is finite when restricted to irreducible manifolds. Its combinatorial nature makes it a computable invariant: using the stratification of the set of 3-manifolds induced by Matveev’s complexity it is possible to produce a list of 3-manifolds up to complexity 10 by means of computer-based computations [Martelli and Petronio 04]. Moreover, the techniques and tools [Matveev 03] that have been set up to study the topology of 3-manifolds in order to produce these lists have allowed the creation of computer programs that “recognize” 3-manifolds [Matveev 05].