Systematic singular triangulations of all orientable Seifert manifolds (Q819523)

Special spines have proved to be a valuable tool for studying \(3\)-manifolds, for instance in complexity theory and normal surface theory [see e.g., \textit{S. Matveev}, Algorithmic topology and classification of 3-manifolds. Algorithms and Computation in Mathematics 9. Berlin: Springer (2003; Zbl 1048.57001) and \textit{B. Martelli} and \textit{C. Petronio}, Geom. Dedicata 108, 15--69 (2004; Zbl 1068.57011)], or for the definition of Turaev-Viro type state sum invariants of 3-manifolds. With a view towards these subjects the authors develop in a systematic way a method for constructing special spines of all orientable Seifert fibered manifolds with orientable base. Knowing that these are obtained by gluing pieces homeomorphic to \((S^2 \setminus \coprod_i D_i^2)\times S^1\), \((S^1 \times S^1 \setminus D^2)\times S^1\) and the \((p,q)\)-type fibered solid torus \(V_{(p,q)}\), the analysis is reduced essentially to the construction of certain special spines \(P_{V_{(p,q)}}\) for the pieces \(V_{(p,q)}\), satisfying the condition that \(P_{V_{(p,q)}}\cap \partial V_{(p,q)}\) is a theta graph with two edges spanning a fiber. An explicit model of \(P_{V_{(p,q)}}\) is given in Section 3, a prefered system of meridian-longitude of \(\partial V_{(p,q)}\) being described in terms of loops embedded in the theta graph. Special spines for the other pieces are considered in Sections 5.1-5.3. Fiber preserving gluing homeomorphisms between the boundary components of all types of pieces are obtained in Section 5.4 by extending the gluings along theta graphs.