Crystallizations of Seifert Fibered 3-Manifolds

Abstract

We study a class of Seifert fibered 3-manifolds M (g,n), depending on two non-negative integers, which arise from polyhedral schemata. Then we completely determine their Seifert invariants by using the crystallization theory, i.e. a representation of closed connected triangulated manifolds by edge-colored graphs. We also obtain a geometric presentation of the fundamental group 7Ti(M(g,n)) corresponding to a spine of M(g,n). Finally we show that M(g, 1) is a 2-fold covering of S 3 branched over a special 3-bridge link. Introduction and notationThroughout this paper we work in the piecewise-linear (PL) category in the sense of [18]. If A" is a finite collection of closed balls, we set \K\ = U{A : A G K}. Then K is said to be a pseudocomplex if and only if 1) \K\ = disjoint U{A : A e K}; 2) If A, B 6 K, then A n B is a union of balls of K; 3) For each /i-ball A £ K, the poset {B 6 K : B C A}, ordered by inclusion, is isomorphic with the lattice of all faces of the standard /i-simplex. The dimension of a pseudocomplex K is defined as max{dim A : A 6 K}. By abuse of language we call simplexes the balls of K and denote by S r (K) the set of all r-simplexes of K. A n-dimensional pseudocomplex K is said to be contracted if the set of its O-simplexes has cardinality n + 1. A pseu-