Complexity Theory of 3-Manifolds

Denote byM the set of all compact 3-manifolds. We wish to study it systematically and comprehensively. The crucial question is the choice of filtration in M. It would be desirable to have a finite number of 3-manifolds in each term of the filtration, all of them being in some sense simpler than those in the subsequent terms. A useful tool here would be a measure of “complexity” of a 3-manifold. Given such a measure, we might hope to enumerate all “simple” manifolds before moving on to more complicated ones. There are several well-known candidates for such a complexity function. For example, take the Heegaard genus g(M), defined to be the minimal genus over all Heegaard decompositions of M. Other examples include the minimal number of simplices in a triangulation of M and the minimal crossing number in a surgery presentation for M.