Representing and recognizing torus bundles over \(\mathbb S^1\) (Q2493398)

The paper belongs to the Italian school (of Mario Pezzana) concerning the representation theory for PL-manifolds of arbitrary dimension via edge-coloured graphs. The author has already written other papers on this subject; in 1999 she gave a complete catalogue \(\widetilde{\mathcal C}^{(26)}\) of nonorientable 3-manifolds admitting colored triangulations up to 26 tetrahedra [Acta Appl. Math. 54, No. 1, 75--97 (1998; Zbl 0913.57011)]. This classification completes the catalogue \({\mathcal C}^{(28)}\) given by \textit{S. Lins} [Gems, computers and attractors for 3-manifolds. Series on Knots and Everything 5, World Scientific, Singapore (1995; Zbl 0868.57002)] and \textit{S. Lins} and \textit{S. Sidki} [Int. J. Algebra and Comput. 5, 205--250 (1995; Zbl 0841.57003)]. Now, the author gives an approach to the study of fiber bundles with base space \(\mathbb{S}^1\) and fiber \(T\) (the torus) via edge-coloured graphs as a combinatorial PL-manifold representation tool. In particular, an algorithmic procedure is described which allows to construct directely from any matrix \(A\in\text{GL}(2;\mathbb{Z})\) a pseudosimplicial triangulation (and, hence, the edge-coloured graph \(\Gamma(A)\) visualizing it) of the torus bundle \(TB(A)\) associated to \(A\). Here \(TB(A)\) is the quotient \((T\times[0, 1])/\sim_A\), where the equivalence relation \(\sim_A\) is given by \((x, 0)\sim_A(\widetilde\phi_A(x),1)\), \(\forall x\in T\), \(\widetilde\phi_A\) being the punctured homeomorphism \((T, x_0)\to(T, x_0)\) \((x_0\in T)\) having \(A\) as an associated matrix. As a consequence, five topologically undetected manifolds of Lins' catalogue whose fundamental group coincides with the fundamental group of a torus bundle are recognized as torus bundles over \(\mathbb{S}^1\). Proposition (5.3). (a) The orientable 3-manifold corresponding to \(r^{28}_{14}\in{\mathcal C}^{(28)}\) (whose fundamental group is \(\mathbb{Z} \left(\begin{smallmatrix} 1 & 0\\ 1 & 1\end{smallmatrix}\right)\)) is the torus bundle \(TB(A)\), with \(A=\left(\begin{smallmatrix} 1 & 0\\ 1 & 1\end{smallmatrix}\right)\); (b) The orientable 3-manifold corresponding to \(r^{28}_5\in{\mathcal C}^{(28)}\) is the torus bundle \(TB(A)\), with \(A= \left(\begin{smallmatrix} 0 & 1\\ -1 & 3\end{smallmatrix}\right)\); (c) The orientable 3-manifold corresponding to \(r^{28}_{10}\in{\mathcal C}^{(28)}\) is the torus bundle \(TB(A)\), with \(A= \left(\begin{smallmatrix} 0 & 1\\ -1 & -3\end{smallmatrix}\right)\); (d) The orientable 3-manifold corresponding to \(r^{28}_{42}\in{\mathcal C}^{(28)}\) is the torus bundle \(TB(A)\), with \(A= \left(\begin{smallmatrix} -1 & 0\\ 2 & -1\end{smallmatrix}\right)\); (e) The orientable 3-manifold corresponding to \(r^{28}_{280}\in{\mathcal C}^{(28)}\) is the torus bundle \(TB(A)\), with \(A= \left(\begin{smallmatrix} 1 & -2\\ 0 & 1\end{smallmatrix}\right)\).