Reidemeister torsion of Seifert fibered spaces for \(SL(2;\mathbb{C})\)-representations
The main result of this paper is the following theorem. Let \(M^3\) denote the orientable Seifert fibered space given by the following Seifert index \(\{b, (\varepsilon, g); (\alpha_1, \beta_1), \dots, (\alpha_m, \beta_m)\}\). Theorem. Let \(\rho : \pi_1 M \to \text{SL} (2; \mathbb{C})\) be an irreducible representation. Then the \(\text{SL} (2; \mathbb{C})\)-torsion \(\tau (M; V_\rho)\) is given by \[ \tau (M; V_\rho) = \begin{cases} 0 \quad & \text{if } H = I \\ 2^{4 - m - 4g} \prod^m_{i = 1} \left( 1 - (-1)^{\nu_i} \cos {\rho_i k_i (\rho) \pi \over \alpha_i} \right) \quad & \text{if } H \neq I,\;\varepsilon = 0 \\ \left( 2 - 2 \cos {s \pi \over N + 1} \right)^{4 - m - 2g} \prod^m_{i = 1} \left( 1 - (-1)^{\nu_i} \cos {\rho_i k_i (\rho) \pi \over \alpha_i} \right) \quad & \text{if } H \neq I,\;\varepsilon = n \end{cases} \] where \(H = \rho (h)\), \(h\) is a representative element of generic fiber in \(\pi_1 M\), \(\rho_i\), \(\nu_i \in \mathbb{Z}\) such that \(|\begin{smallmatrix} \alpha_i & \rho_i \\ \beta_i & \nu_i \end{smallmatrix} |= - 1\) and \(0 < \rho_i < \alpha_i\), \(k_i (\rho) \in \mathbb{Z}\) such that \(0 \leq k_i \leq \alpha_i\), and \(k_i (\rho) \equiv \beta_i \bmod 2\), \(N = \beta_1/ \alpha_1 + \cdots + \beta_m/ \alpha_m\), \(s \in \mathbb{Z}\) such that \(0 \leq s \leq 2N + 2\). (1) In general the dimension of the space of representations of a Seifert fibered space is not zero; in particular the distinct classes of irreducible representations are not finite. However the set of the \(\text{SL} (2; \mathbb{C})\)-torsion turns out to be a finite subset in \(\mathbb{R}\) by this theorem; that is \(\text{SL} (2; \mathbb{C})\)-torsion is a constant function on each connected component of the space of irreducible representations. (2) It may be a problem to determine whether there exists a 3-manifold with continuous variations of the \(\text{SL} (2; \mathbb{C})\)-torsion. In fact the answer is yes. In our paper [Reidemeister torsion of the figure-eight knot exterior for \(\text{SL} (2; \mathbb{C})\)-representations, Osaka Math. J. 31, No. 3, 523-532 (1994)], we will prove that the double of the figure-eight knot exterior in \(S^3\) has continuous variations of the \(\text{SL} (2; \mathbb{C})\)-torsion.
scientific article; zbMATH DE number 645685