Non-commutative multivariable Reidemester torsion and the Thurston norm.

*(English)*Zbl 1147.57014S. Harvey introduced in [Topology 44, No. 5, 895–945 (2005; Zbl 1080.57019)] a function \(\delta_n\) on the first cohomology group of a manifold, that gives the degree of the \(n\)-th order Alexander polynomial, defined from the rational derived series. On the other hand, in [Pac. J. Math. 230, No. 2, 271–296 (2007)], S. Friedl had related these higher order Alexander polynomials to Reidemeister torsions. This relationship is analyzed here and used to show that \(\delta_n\) is a seminorm.

The key tool here is to consider the natural representation from \(\pi_1(M)\) to \(K[t_1^{\pm 1},\cdots t_m^{\pm 1}]\), namely the ring of Laurent polynomials on \(m\) conmuting variables over an skew field \(K\), where \(m\) is the first Betti number of \(M\), \(K\) is defined from the group ring of \(\pi/\pi^n\) localized by \(\pi^1/\pi^n\), and \(\pi^n\) denotes the \(n\)-th term of the rational derived series. This representation is acyclic and, by naturality, its Reidemeister torsion contains the information of the \(n\)-th Alexander polynomial corresponding to any element of \(H^1(M;\mathbb Z)\). All these constructions need a careful use of noncomutative group and ring theory, and using them, the authors can prove that \(\delta_n\) is a seminorm on \(H^1(M,\mathbb Z)\).

As a corollary, the authors show that these invariants stabilize: for any 3-manifold there exists \(n_0\) so that \(\delta_n=\delta_{n_0}\) for all \(n > n_0\). The paper ends with some examples that illustrate the use of this result to compute the Thurston norm on some links, because it is bounded below by Harvey’s invariants.

The key tool here is to consider the natural representation from \(\pi_1(M)\) to \(K[t_1^{\pm 1},\cdots t_m^{\pm 1}]\), namely the ring of Laurent polynomials on \(m\) conmuting variables over an skew field \(K\), where \(m\) is the first Betti number of \(M\), \(K\) is defined from the group ring of \(\pi/\pi^n\) localized by \(\pi^1/\pi^n\), and \(\pi^n\) denotes the \(n\)-th term of the rational derived series. This representation is acyclic and, by naturality, its Reidemeister torsion contains the information of the \(n\)-th Alexander polynomial corresponding to any element of \(H^1(M;\mathbb Z)\). All these constructions need a careful use of noncomutative group and ring theory, and using them, the authors can prove that \(\delta_n\) is a seminorm on \(H^1(M,\mathbb Z)\).

As a corollary, the authors show that these invariants stabilize: for any 3-manifold there exists \(n_0\) so that \(\delta_n=\delta_{n_0}\) for all \(n > n_0\). The paper ends with some examples that illustrate the use of this result to compute the Thurston norm on some links, because it is bounded below by Harvey’s invariants.

Reviewer: Joan Porti (Bellaterra)

##### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

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\textit{S. Harvey} and \textit{S. Friedl}, Algebr. Geom. Topol. 7, 755--777 (2007; Zbl 1147.57014)

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