Not finished yet…
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
Let us first unravel what this theorem really means step by step.
The manifold \(M\)
Lets inspect the 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.
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”
For this we need to define the 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:
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):
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.
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:
As an example of a torus which is isotopic to a boundary component is given here (for \(\partial M \cong \mathbb{T}\))
Seifert manifolds
Specifically, these are Seifert manifolds:
and these are not seifert manifolds:
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
Because \(\text{Spin}^c(V)\) is the double cover of Lie groups1 of \(\text{SO}(V) \times U_1\), we can use these to describe spinor bundles.
Footnotes
The \(L^2\)-torsion function and the Thurston norm of 3-manifolds - Bohn - 2007 - An introduction to Seiberg-Witten theory on closed 3-manifolds.pdf, p158↩︎