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Most of the screenshots of definitions/theorems are taken from:

Introduction to 3-manifolds by Wolfgang Lück at Bonn Germany

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:

The main theorem

Theorem 5.1 (Main theorem). Let $M$ be an irreducible 3-manifold with infinite fundamental group $\pi$ which is not a closed graph manifold and not homeomorphic to $S^1 \times D^2$. Let $\mathfrak{s} \in \mathrm{Spin}^c(M)$ and write $\pi = \pi_1(M)$.

Then there exists a $(H_1)_f$-factorizing epimorphism $\alpha\colon \pi \to \Gamma$ to a virtually finitely generated free abelian group such that the following holds: For any $\phi \in H^1(M; \mathbb{Q})$ and any factorization of $\alpha\colon \pi \to \Gamma$ into group homomorphisms $\pi \xrightarrow{\mu} G \xrightarrow{\nu} \Gamma$ for a residually finite countable group $G$, there exists a real number $D$ depending only on $\phi$ but not on $\mu$ such that for $t \leq 1$

\[\tfrac{1}{2}\bigl(\phi(c_1(\mathfrak{s})) + x_M(\phi)\bigr)\ln t - D \;\leq\; \rho^{(2)}(M, \mathfrak{s}; \mu, \phi)(t) \;\leq\; \tfrac{1}{2}\bigl(\phi(c_1(\mathfrak{s})) + x_M(\phi)\bigr)\ln t\]

and such that for $t \geq 1$

\[\tfrac{1}{2}\bigl(\phi(c_1(\mathfrak{s})) - x_M(\phi)\bigr)\ln t - D \;\leq\; \rho^{(2)}(M, \mathfrak{s}; \mu, \phi)(t) \;\leq\; \tfrac{1}{2}\bigl(\phi(c_1(\mathfrak{s})) - x_M(\phi)\bigr)\ln t.\]

In particular we get

\[\deg\bigl(\rho(M, \mathfrak{s}; \mu, \phi)\bigr) = -x_M(\phi).\]

Let us first unravel what this theorem really means step by step. Here are the terms we need to understand:

The 3-manifold $M$ is
- irreducible
- has infinite fundamental group
- is not a closed graph manifold
- is not homeomorphic to $S^1 \times D^2$
$\mathfrak{s} \in \text{Spin}^c(M)$ is a spin structure on $M$.
$\alpha\colon \pi \to \Gamma$
- is an epimorphism (surjective group homomorphism)
- is $(H_1)_f$-factorizing, which means that $\alpha$ factors through the free part of the first homology group $H_1(M)_f $
- $\Gamma$ is a virtually finitely generated free abelian group
$\phi \in H^1(M; \mathbb{Q})$ is a rational cohomology class on $M$.
$\mu : \pi \to G$ is a group homomorphism to a residually finite countable group $G$.
- A group $G$ is residually finite (aka finitely approximable) if for every non-identity element $g \in G$, there exists a finite index normal subgroup $N \triangleleft G$ such that $g \notin N$. Intuitively, this means that the group can be approximated by its finite quotients.
$\rho^{(2)}(M, \mathfrak{s}; \mu, \phi)(t)$ is the $L^2$-torsion function associated to $M$, $\mathfrak{s}$, $\mu$, and $\phi$ evaluated at $t$.

The manifold $M$

Lets inspect the first requirements:

Let $M$ be an irreducible 3-manifold with infinite fundamental group $\pi$ which is not a closed graph manifold and not homeomorphic to $S^1 \times D^2$. Let $\mathfrak{s} \in \mathrm{Spin}^c(M)$ and write $\pi = \pi_1(M)$.

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.

Irreducible

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

The prime decomposition theorem A 3-manifold $N$ is called prime if $N$ can not be written as a non-trivial connected sum of two manifolds, i.e. if $N = N_1 \# N_2$, then $N_1 = S^3$ or $N_2 = S^3$. Furthermore $N$ is called irreducible if every embedded $S^2$ bounds a 3-ball. Note that an irreducible 3-manifold is prime, conversely if $N$ is a prime 3-manifold, then either $N$ is irreducible or $N = S^1 \times S^2$. We now have the following theorem:

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

Given 2-oriented 3-manifolds $N_1$ and $N_2$ we can consider the connected sum \[N_1 \# N_2 = (N_1 \setminus \text{3-ball}) \cup (N_2 \setminus \text{3-ball}),\] where we identify the two boundary spheres using an orientation reversing homeomorphism. Note that the diffeomorphism type of the

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

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)

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

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

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.

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.

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:

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:

Two tori which are embedded in a cube with two cylinders removed which are not disjoint, but which are both not isotopic to the single boundary component — Two tori which are embedded in a cube with two cylinders removed which are *not* disjoint, but which are both *not* isotopic to the single boundary component

A finite family of disjoint, pairwise-nonisotopic incompressible tori in M which are not isotopic to boundary components — A finite family of disjoint, pairwise-nonisotopic incompressible tori in $M$ which are not isotopic to boundary components

Clearly both of these tori clearly cannot be isotopic to the boundary

By inflating each torus, we are able to completely cover M — By inflating each torus, we are able to completely cover $M$

As an example of a torus which is isotopic to a boundary component is given here (for $\partial M \cong \mathbb{T}$)

A torus which is isomorphic to a boundary component of M — A torus which is isomorphic to a boundary component of $M$

Seifert manifolds

seifert manifold are 3-manifolds with one of these geometries (a metric tensor)

Specifically, these are Seifert manifolds:

S^2 \times \mathbb{R} see video — $S^2 \times \mathbb{R}$ see video

\mathbb{H}^2 \times \mathbb{R} see video — $\mathbb{H}^2 \times \mathbb{R}$ see video

\text{Nil} see video — $\text{Nil}$ see video

\widetilde{\text{Sl}(2, \mathbb{R})} see video — $\widetilde{\text{Sl}(2, \mathbb{R})}$ see video

and these are not seifert manifolds:

\mathbb{H3} see video — $\mathbb{H3}$ see video

\text{Solv} see video — $\text{Solv}$ see video

geometrically atoroidal

closed

A manifold without bounadary which is compact is called closed, so $M$ must have empty boundary.

Graph manifolds

By Wolfang Lück’s talk as mentioned above, we have this definition:

And we have already inspected this under infinite-fundamental-group

the spin structure $\mathfrak{s} \in \text{Spin}^c(M)$

We will not look too much into what this means. By Liu we know that omitting this parameter in the function $\rho$ from the main theorem, we still obtain the same asymptotic degree. In fact, omitting this parameter yields a family of functions $[\rho] = \{\rho(t) \cdot t^r \mid \exists r \in \mathbb{R}\}$

This however does not change $\deg(\rho)$. For clarity though, we have

Spin and Spin^c. Here $(V, g)$ denotes a real vector space $V$ equipped with an inner product $g$, i.e., a symmetric, positive-definite bilinear form. In the setting of 3-manifolds, $V$ is typically a tangent space $T_p M$ and $g$ the Riemannian metric.

Recall that the Spin group associated to $(V, g)$ is defined by \[\operatorname{Spin}(V) := \{ v_1 \cdots v_m \mid m \text{ even}, |v_i| = 1 \} \subset \operatorname{Cl}(V)^*,\] where $\operatorname{Cl}(V)^*$ denotes the group of units in the Clifford algebra $\operatorname{Cl}(V)$. More generally, given a vector space $V$ over a field $K$ and a quadratic form $Q : V \to K$, the Clifford algebra is defined as the quotient algebra \[\operatorname{Cl}(V, Q) = T(V)/I_Q,\] where $I_Q$ is the two-sided ideal in the tensor algebra $T(V)$ generated by all elements of the form $v \otimes v - Q(v)1$ for all $v \in V$.

Definition (Complex Spin Group, D 1.1). The complex spin group $\operatorname{Spin}^c(V)$ is the group generated by $\operatorname{Spin}(V)$ and $\mathrm{U}_1$ inside the group $\operatorname{Cl}^c(V)^*$. If $V = \mathbb{R}^n$, we write $\operatorname{Spin}^c_n := \operatorname{Spin}^c(\mathbb{R}^n)$.

Since $\mathrm{U}_1$ lies in the center of $\operatorname{Cl}^c(V)$ and $\operatorname{Spin}(V) \cap \mathrm{U}_1 = \{\pm 1\}$, it follows that \[\operatorname{Spin}^c(V) = \operatorname{Spin}(V) \times_{\mathbb{Z}_2} \mathrm{U}_1.\]

Definition (Spin^c Structure, D 2.2). A spin^c structure on $M$, denoted by $\sigma$, consists of a principal $\operatorname{Spin}^c_n$-bundle $P_{\operatorname{Spin}^c}(\sigma)$ together with a bundle map \[\xi^c : P_{\operatorname{Spin}^c}(\sigma) \to P_{\mathrm{SO}}\] which is $\operatorname{Spin}^c_n$-equivariant with respect to the homomorphism $\xi^c_0 : \operatorname{Spin}^c_n \to \mathrm{SO}_n$. The pair $(M, \sigma)$ is called a spin^c manifold.

Since $\operatorname{Spin}^c(V)$ is a double cover of Lie groups of $\operatorname{SO}(V) \times \mathrm{U}_1$, these structures can be used to describe spinor bundles.

Spinc principal bundle over M — Spin^c principal bundle over $M$

The first Chern class. The key datum attached to a Spin^c structure $\mathfrak{s}$ that appears in Theorem 5.1 is its first Chern class $c_1(\mathfrak{s}) \in H^2(M; \mathbb{Z})$. Given $\mathfrak{s}$, one forms the determinant line bundle \[L(\mathfrak{s}) := P_{\operatorname{Spin}^c}(\mathfrak{s}) \times_{\operatorname{Spin}^c_3} \mathbb{C},\] where $\operatorname{Spin}^c_3$ acts on $\mathbb{C}$ via the determinant character $[q, z] \mapsto z^2$; then $c_1(\mathfrak{s}) := c_1(L(\mathfrak{s})) \in H^2(M;\mathbb{Z})$.

For a closed oriented 3-manifold $M$, Spin^c structures always exist (since $w_2(M) = 0$ for all orientable 3-manifolds, one can always lift the frame bundle from $\operatorname{SO}_3$ to $\operatorname{Spin}^c_3$). Moreover, the set $\operatorname{Spin}^c(M)$ is an affine torsor over $H^2(M;\mathbb{Z})$: given two Spin^c structures $\mathfrak{s}, \mathfrak{s}'$ there is a unique $a \in H^2(M;\mathbb{Z})$ such that $c_1(\mathfrak{s}') = c_1(\mathfrak{s}) + 2a$.

In the statement of Theorem 5.1, the Spin^c class enters through the combination $\phi(c_1(\mathfrak{s}))$, which for a closed oriented 3-manifold is the integer \[\phi(c_1(\mathfrak{s})) = \langle \phi \cup c_1(\mathfrak{s}),\, [M] \rangle \in \mathbb{Z},\] where $\phi \in H^1(M;\mathbb{Z})$, $c_1(\mathfrak{s}) \in H^2(M;\mathbb{Z})$, and $[M] \in H_3(M;\mathbb{Z})$ is the fundamental class. This shifts the asymptotic slopes of $\rho^{(2)}$ by a fixed amount depending on $\mathfrak{s}$; in particular, the degree $\deg(\rho) = -x_M(\phi)$ is independent of $\mathfrak{s}$.

TODO

\[\boxed{\deg\bigl(\rho^{(2)}(M,\mathfrak{s};\mu,\phi)\bigr) = -x_M(\phi) \quad \text{(up to the Spin}^c\text{ shift)}}\]

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

The main theorem

The manifold \(M\)

Irreducible

Infinite fundamental group

Seifert manifolds

geometrically atoroidal

closed

Graph manifolds

the spin structure \(\mathfrak{s} \in \text{Spin}^c(M)\)

TODO