[Paper Review] The $L^2$-Alexander torsion for Seifert fiber spaces

Gerrit Herrmann|arXiv (Cornell University)|Feb 28, 2016

Geometric and Algebraic Topology13 references3 citations

TL;DR

This paper computes the $L^2$-Alexander torsion for Seifert fiber spaces and graph manifolds, showing it is completely determined by the Thurston norm. Using $S^1$-actions and orbifold Euler characteristics, the authors prove that $\tau^{(2)}(M,\phi,\gamma) \doteq \max\{1, t^{x_M(\phi)}\}$, establishing a precise link between analytic torsion and geometric topology invariants for these 3-manifolds.

ABSTRACT

We calculate the $L^2$-Alexander torsion for Seifert fiber spaces and graph manifolds in terms of the Thurston norm.

Motivation & Objective

• To determine the $L^2$-Alexander torsion for Seifert fiber spaces in terms of geometric invariants.
• To extend this computation to graph manifolds under specific conditions on the homomorphism $\gamma$.
• To establish a precise relationship between the $L^2$-Alexander torsion and the Thurston norm $x_M(\phi)$.
• To provide a uniform formula for the torsion that captures its dependence on the minimal complexity of surfaces dual to $\phi$.
• To resolve a conjecture on the $L^2$-Alexander torsion for knot complements in $S^3$ using this framework.

Proposed method

• Use of $S^1$-CW-complexes with effective $S^1$-actions to model Seifert fiber spaces with orientable base orbifolds.
• Application of Lemma B, which expresses $\tau^{(2)}(X,\phi,\gamma)$ in terms of the $S^1$-orbifold Euler characteristic $\chi^{S^1}_{\text{orb}}(X)$ and the pairing $k_\phi = \phi([F])$.
• Proof of Lemma A, relating the Thurston norm $x_M(\phi)$ to the $S^1$-orbifold Euler characteristic and the pairing $k_\phi$.
• Use of the gluing formula for the Thurston norm (Theorem 3.5) to decompose non-orientable base spaces into orientable pieces.
• Application of doubling arguments for Möbius strips to reduce to the Klein bottle case, which is handled via Lemma 3.6.
• Use of the equivalence relation $f \doteq g$ to simplify the torsion expression to $\max\{1, t^{x_M(\phi)}\}$.

Experimental results

Research questions

• RQ1 How is the $L^2$-Alexander torsion of a Seifert fiber space related to its Thurston norm?
• RQ2 Can the $L^2$-Alexander torsion be computed uniformly across all Seifert fiber spaces using geometric data?
• RQ3 What role does the $S^1$-action play in computing the $L^2$-Alexander torsion for manifolds with fibered structure?
• RQ4 Does the $L^2$-Alexander torsion of a graph manifold depend only on the Thurston norm of the cohomology class?
• RQ5 Can the torsion formula be extended to non-orientable base spaces of Seifert fibered spaces?

Key findings

• The $L^2$-Alexander torsion for a Seifert fiber space $M$ (excluding $S^1 \times S^2$ and $S^1 \times D^2$) satisfies $\tau^{(2)}(M,\phi,\gamma) \doteq \max\{1, t^{x_M(\phi)}\}$ when the image of a regular fiber under $\gamma$ has infinite order.
• For graph manifolds where each JSJ component satisfies the same condition on $\gamma$, the torsion is also $\max\{1, t^{x_M(\phi)}\}$, extending the result beyond Seifert spaces.
• The proof relies on Lemma A, which establishes $x_M(\phi) = |\chi^{S^1}_{\text{orb}}(M) \cdot k_\phi|$ for $k_\phi = \phi([F])$.
• Lemma B provides a general formula: $\tau^{(2)}(X,\phi,\gamma) \doteq \max\{1, t^{k_\phi}\}^{-\chi^{S^1}_{\text{orb}}(X)}$ for $S^1$-CW-complexes with injective $\gamma \circ ev_x$.
• The result holds for non-orientable base spaces by cutting into orientable pieces (Klein bottles and Möbius strips), using gluing and doubling techniques.
• The formula $\tau^{(2)}(S^3 \setminus \nu K, \phi_K, \text{id}) \doteq \max\{1, t\}^{-1}$ is recovered, showing the knot is trivial iff the torsion is $\max\{1, t\}^{-1}$.