$$ \usepackage{amssymb} \newcommand{\N}{\mathbb{N}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ZZ}{\ooalign{Z\cr\hidewidth\kern0.1em\raisebox{-0.5ex}{Z}\hidewidth\cr}} \newcommand{\colim}{\text{colim}} \newcommand{\weaktopo}{\tau_\text{weak}} \newcommand{\strongtopo}{\tau_\text{strong}} \newcommand{\normtopo}{\tau_\text{norm}} \newcommand{\green}[1]{\textcolor{ForestGreen}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} \newcommand{\orange}[1]{\textcolor{orange}{#1}} \newcommand{\tr}{\text{tr}} \newcommand{\id}{\text{id}} \newcommand{\im}{\text{im}\>} \newcommand{\res}{\text{res}} \newcommand{\TopTwo}{\underline{\text{Top}^{(2)}}} \newcommand{\CW}[1]{\underline{#1\text{-CW}}} \newcommand{\ZZ}{% \ooalign{Z\cr\hidewidth\raisebox{-0.5ex}{Z}\hidewidth\cr}% } % specific for this document \newcommand{\cellOne}{\textcolor{green}{1}} \newcommand{\cellTwo}{\textcolor{red}{2}} \newcommand{\cellThree}{\textcolor{brown}{3}} \newcommand{\cellFour}{\textcolor{YellowOrange}{4}} $$

The \(\ell^2\)-Alexander torsion and the Thurston norm

algebraic topology
3-manifolds
Author

Luca Leon Happel

Published

February 7, 2026

Warning

Not finished yet…

My masters thesis is about the following statement from the 2019 paper “The \(L^2\)-torsion function and the Thurston norm of 3-manifolds” by Friedl and Lück:

Main Theorem

Main Theorem

Let us first unravel what this theorem really means step by step.

The manifold \(M\)

Lets inspect the first requirements:

first requirements

first requirements

Further up in the text the authors specify that all manifolds are compact, connected, and oriented, unless otherwise specified. So \(M\) must be as well.

M must be compact, connected, oriented

\(M\) must be compact, connected, oriented

We will also assume like Friedl and Lück that all 3-manifolds are compact, connected, and oriented, unless otherwise specified.

Irreducible

We have a definition by Friedl in “AN INTRODUCTION TO 3-MANIFOLDS”

Definiton for M being prime/irreducible

Definiton for \(M\) being prime/irreducible

For this we need to define the connected sum of two 3-manifolds:

connected sum of two 3-manifolds

connected sum of two 3-manifolds

In the case of 2-manifolds, this video visualizes it well:

and for 3-manifolds (specifically \(\mathbb{R}^3 \# \mathbb{R}^3\)) this can be seen as adding a wormhole between two copies of \(\mathbb{R}^3\):

Or another way to look at this connected sum is by “cutting out” some region in \(N_1\) and “pasting” \(N_2\) in there. We may even do this an infinite amount of time in a checkerboard pattern, as shown here:

In this video the relevant section shows this infinite tiling where 3-manifolds with different geometries are glued together in an alternating pattern:

Screenshot from “Portals to Non-Euclidean Worlds” by Tehora Rogue

Screenshot from “Portals to Non-Euclidean Worlds” by Tehora Rogue

So being irreducible means either of the following equivalent conditions hold: - every embedded \(S^2\) of \(M\) bounds a \(D^3\). - \(M\) is prime (\(M = N_1\# N_2 \implies N_1 = S^3 \text{ or } N_2 = S^3\)) or \(M = S^1 \times S^2\).

Infinite fundamental group

Aka. \(\|\pi_1(M)\| = \infty\). We use this to apply the famous Thurston geometrization theorem (these slides are from Lücks “Introduction to 3-manifolds” talk in Bonn):

Thurston’s geometrization theorem (formerly conjecture, but famously proved by Perelmann)

Thurston’s geometrization theorem (formerly conjecture, but famously proved by Perelmann)

What is a “geometric toral splitting” you may ask?

theorm of uniqueness

theorm of uniqueness

geometric toral splitting

geometric toral splitting

Here is a tool to play around this this definition (it only includes tori, and no framed knots!):

The individual tori above are have a globe texture each:

geometric toral splitting

geometric toral splitting

A surface \(\iota : S \hookrightarrow M\) is incompressible, if it induces an injection on the fundamental groups, i.e. \(\pi_1(\iota) : \pi_1(S) \hookrightarrow \pi_1(M)\) is injective.

JSJ Step 1

JSJ Step 1

What could cause \(\pi_1(\iota)\) to fail at being injective? By the Hurewicz theorem, the first homology group \(H_1(S)\) is the abelianization of \(\pi_1(S)\), but for \(S\) a torus, \(\pi_1(S) = \mathbb{Z}^2\) is already abelian, so \(H_1(S) = \pi_1(S)\). \(H_1(S)\) is generated by the longitude and the meridian of the torus, so if either of those is sent to zero by \(\pi_1\), then \(\pi_1(\iota)\) cannot be injective.

JSJ Step 2

JSJ Step 2

Remember, that either of these loops will dissapear, if the ambient manifold \(M\) contains a disk (a compressing disk), which will allow these loops to be contracted to a point:

JSJ Step 3 (meridian)

JSJ Step 3 (meridian)

JSJ Step 3 (longitude)

JSJ Step 3 (longitude)

Well, these disks cannot exist, iff \(M\) contains some obstruction to prevent such complressing disks.

JSJ Step 4 (example of two possible such obstructions. The grey tubes are ment to symbolize space removed from \mathbb{R}^3)

JSJ Step 4 (example of two possible such obstructions. The grey tubes are ment to symbolize space removed from \(\mathbb{R}^3\))

This also explains what it means for \(M\) to have “incompressible boundary”: \[ \pi_1(\partial M) \hookrightarrow \pi_1(M) \text{ is injective} \]

This covers “disjoint” and “incompressible”. What does it mean for a torus to not be isotopic to a boundary component? Lets consider an example: