Additivity of complexity and Haken's method in the topology of three- dimensional manifolds
Denote by \({\mathcal M}\) the set of all compact 3-dimensional manifolds (homeomorphic manifolds are considered the same), and by \(\Delta\) the underlying polyhedron of the 1-skeleton of the standard 3-simplex, i.e. a polyhedron homeomorphic to the circle with three radii. A polyhedron P is called almost special if the link of each point of P can be embedded into \(\Delta\). A function d: \({\mathcal M}\to {\mathbb{Z}}_+\) is defined as follows: \(d(M)=k\in {\mathbb{Z}}_+\) if M has an almost special spine which has k vertices and does not have an almost special spine with less than k vertices. It is proved that the number k(M), which is called the complexity of M, has the following properties: (a) \(d^{-1}(k)\) contains finitely many closed irreducible manifolds (finiteness); (b) \(d(M_ 1\#M_ 2)=d(M_ 1)+d(M_ 2)\), where \(M_ 1\#M_ 2\) denotes the connected sum of \(M_ 1\) and \(M_ 2\) (additivity). The proof of (a) is short and based on previous results of the author, while the proof of (b) is based on \textit{W. Haken}'s theory of surfaces [Theorie der Normalflächen. Ein Isotopiekriterium für den Kreisknoten. Acta Math. 105, 245-375 (1961; Zbl 0100.194)].
scientific article; zbMATH DE number 4143049