Complexity of 3-manifolds

Summary

Abstract

We give a summary of known results on Matveev's complexity of compact 3-manifolds. The only relevant new result is the classification of all closed orientable irreducible 3-manifolds of complexity 10.

Introduction

In 3-dimensional topology, various quantities are defined, that measure how complicated a compact 3-manifold M is. Among them, we find the Heegaard genus, the minimum number of tetrahedra in a triangulation, and Gromov's norm (which equals the volume when M is hyperbolic). Both Heegaard genus and Gromov norm are additive on connected sums, and behave well with respect to other common cut-and-paste operations, but it is hard to classify all manifolds with a given genus or norm. On the other hand, triangulations with n tetrahedra are more suitable for computational purposes, since they are finite in number and can be easily listed using a computer, but the minimum number of tetrahedra is a quantity which does not behave well with any cut-and-paste operation on 3-manifolds. (Moreover, it is not clear what is meant by “triangulation”: do the tetrahedra need to be embedded? Are ideal vertices admitted when M has boundary?)

In 1988, Matveev introduced [Mat88] for any compact 3-manifold M a non-negative integer c(M), which he called the complexity of M, defined as the minimum number of vertices of a simple spine of M. The function c is finite-to-one on the most interesting sets of compact 3-manifolds, and it behaves well with respect to the most important cut-and-paste operations.