On the stable complexity and the stable presentation length for 3-manifolds (Intelligence of Low-dimensional Topology)
建一 吉田 · Kyoto University Research Information Repository (Kyoto University) · 2015
IntroductionThis article is a survey of the stable complexity introduced by Francaviglia, FYigerio, and Martelli [4] and the stable presentation length introduced by Yoshida [19].We will consider some invariants for a 3-manifold.We assume that a 3-manifold is oriented, compact, and possibly with boundary consisting of tori, unless otherwise stated.We define a finite volume hyperbolic 3-manifold to be a compact 3-manifold whose interior admits a complete metric of constant sectional curvature $-1$ and finite volume, Perelman [14,15] proved the geometrization of a 3-manifold.A closed 3-manifold admits the prime decomposition, i.e. the maximal decomposition by connected sums.After performing the prime decomposition, each component is an irreducible manifold or $S^{1}\cross S^{2}.$An irreducible 3-manifold admits the JSJ decomposition, which is a decomposition along essential tori.The geometrization implies that each piece after the JSJ decomposition is a Seifert fibered manifold or a finite volume hyperbolic manifold.Milnor and Thurston [12] considered some characteristic numbers of manifolds.An in- variant $C$ of manifolds is a characteristic number if $C(N)=d\cdot C(M)$ for any $d$ -sheeted covering $Narrow M$ .For example, Milnor and Thurston introduced the following characteristic number, which is called the stable $\triangle$ -complexity by Fkancaviglia, Frigerio, and Martelli [4].where the infimum is taken for the finite sheeted coverings of $M$ .The stable $\triangle$ -complexity of a 3-manifold is almost same as the stable complexity.Gromov [5] introduced the simplicial volume of a manifold, and showed that the sim- plicial volume of a hyperbolic manifold is proportional to its volume.In particular, the