A New Decomposition Theorem for 3-Manifolds
نویسنده
چکیده
Introduction We develop in this paper a theory of complexity for pairs (M,X) where M is a compact 3-manifold such that χ(M) = 0, and X is a collection of trivalent graphs, each graph τ being embedded in one component C of ∂M so that C \ τ is one disc. In the special case where M is closed, so X = ∅, our complexity coincides with Matveev’s [6]. Extending his results we show that complexity of pairs is additive under connected sum and that, whenM is closed, irreducible, P-irreducible and different from S, L3,1,P , its complexity is precisely the minimal number of tetrahedra in a triangulation. These two facts show that indeed complexity is a very natural measure of how complicated a manifold or pair is. The former fact was known to Matveev in the closed case, the latter one in the orientable case. The most relevant feature of our theory is that it leads to a splitting theorem along tori and Klein bottles for irreducible and P-irreducible pairs (so, in particular,
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