A New Decomposition Theorem for 3Manifolds
Abstract
Let M be a (possibly nonorientable) compact 3manifold with (possibly empty) boundary consisting of tori and Klein bottles. Let $X\subset\partial M$ be a trivalent graph such that $\partial M\setminus X$ is a union of one disc for each component of $\partial M$. Building on previous work of Matveev, we define for the pair (M,X) a complexity c(M,X) and show that, when M is closed, irreducible and P^2irreducible, $c(M,\emptyset)$ is the minimal number of tetrahedra in a triangulation of M. Moreover c is additive under connected sum, and, given any n>=0, there are only finitely many irreducible and P^2irreducible closed manifolds having complexity up to n. We prove that every irreducible and P^2irreducible pair (M,X) has a finite splitting along tori and Klein bottles into pairs having the same properties, and complexity is additive on this splitting. As opposed to the JSJ decomposition, our splitting is not canonical, but it involves much easier blocks than all Seifert and simple manifolds. In particular, most Seifert and hyperbolic manifolds appear to have nontrivial splitting. In addition, a given set of blocks can be combined to give only a finite number of pairs (M,X). Our splitting theorem provides the theoretical background for an algorithm which classifies 3manifolds of any given complexity. This algorithm has been already implemented and proved effective in the orientable case for complexity up to 9.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2001
 arXiv:
 arXiv:math/0105034
 Bibcode:
 2001math......5034M
 Keywords:

 Geometric Topology;
 57M20;
 57N10
 EPrint:
 32 pages, 16 figures