A new decomposition theorem for 3-manifolds

Abstract

Let $M$ be a (possibly non-orientable) compact 3-manifold 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 $c(M,X)\in\mathbb{N}$ and show that, when $M$ is closed, irreducible and $\mathbb{P}^2$-irreducible, $c(M,\emptyset)$ is the minimal number of tetrahedra in a triangulation of $M$. Moreover $c$ is additive under connected sum, and, given any $n\geqslant 0$, there are only finitely many irreducible and $\mathbb{P}^2$-irreducible closed manifolds having complexity up to $n$. We prove that every irreducible and $\mathbb{P}^2$-irreducible 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 non-trivial 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 3-manifolds of any given complexity. This algorithm has been already implemented and proved effective in the orientable case for complexity up to 9.