Complexity theory of three-dimensional manifolds. (English) Zbl 0724.57012

Each compact 3-manifold has a spine which is almost special - a polyhedron in which the link of each vertex embeds in the 1-skeleton of the 3-simplex. A complexity is defined for 3-manifolds by taking the minimum over all almost special spines of the number of vertices for which the link is the entire 1-skeleton of the 3-simplex. Some properties of this complexity are established: (A) there are only finitely many 3- manifolds with a given complexity, (B) complexity is additive over connected sums, and (C) cutting an irreducible, \(\partial\)-irreducible 3- manifold along an incompressible, and \(\partial\)-incompressible surface cannot increase complexity. A list of all 3-manifolds with complexity \(\leq 6\) is given.
Reviewer: J.Hempel (Houston)


57N10 Topology of general \(3\)-manifolds (MSC2010)
57Nxx Topological manifolds