Complexity Theory of 3-Manifolds

  • Sergei Matveev
Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 9)


Denote by M 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 sim-plices in a triangulation of M and the minimal crossing number in a surgery presentation for M.


Boundary Curve Lens Space Solid Torus Regular Neighborhood Special Spine 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Sergei Matveev
    • 1
  1. 1.Chelyabinsk State UniversityChelyabinskRussia

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