Bena Tshishiku - Surface bundles

Abstract

Possibly: expository talk on surface bundles to prepare for later talks. The main question is how can we classify/distingish surface bundles?

Talk

We define a Surface bundle like this $S_{g,n}\to E\to B$ which is something that locally looks like a product. $B$ will be a circle or a Surface.

Example

Taking the $3$-sphere and removing the Kleeblattschlinge gives us a surface bundle. The trefoil can be described by $z^2=w^3$ in complexly parametrized $S^3$.

We draw the Seifert-Fläche of the knot. This gives us a fiber of the bundle.

Example

Taking $S^3$ and removing the figure eight knot. Drawing in the Seifert surface gives us the same fiber and the same base but the resulting fiber bundles are different!

Note that the base is the circle. Such spaces are usually not hard to understand.

Theorem

Every surface bundle over the circle is a Mapping torus. Two mapping tori are isomorphic iff. the underlying mapping classes are in the same conjugacy class.

“Monodromy” of the bundles are matrices. I wonder how they’re defined.

Monodromy classification

Theorem

When the Euler characteristic of $S_{g,n}$ is negative then the bundles $S_{g,n}\to E\to B$ are one-to-one to homeomorphisms from ${\pi }_1(B)$ into $MCG(S)$ (up to conjugacy classes).

This uses the fact that ${\pi }_1(B)$ consists of circles, giving us a Bundle over a circle which can be classified with a mapping torus. Remember, that such homeomorphisms are not weird. Point pushes map the fundamental group into the Mapping class group.

This means: Given ${\pi }_1(S_h)\subseteq MCG(S_g)$ we can build a surface bundle. But this construction doesn’t need to be obvious. It is easy, if the homomorphism factors through $\text{Diff}(S_g)$.

Theme: We access the geometry or topology of surface bundles using the Mapping class group acting on Teichmüller-space. Question: What Geometry does $E$ admit and how does it depend on the projection map $p_E$.

$\varphi$	Geometry of $M(\varphi )$
finite order	${\mathbb{H}}^2\times \mathbb{R}$ geometry (Nielsen)
reducible	Has Nontrivial JSJ-decomposition (contains incompressible torus in $M(\varphi )$)
Pseudo-Anosov Homeomorphism	${\mathbb{H}}^3$ geometry (Thurston)

We sketch Nielsen’s result. Given finite order class $[\varphi ]$. We want to get a concrete Diffeomorphismus $\psi$ where the power is equal to the identity. Then we want to construct a hyperbolic metric on $S$ which is acted open by $\psi$ as an isometry.

For that we use the Teichmüller-space. Since the Teichmüller spacce parametrises hyperbolic geometries this becomes a fixed point problem (find a point which is fixed by $\varphi$). For that we use that the Teichmüller space is homeomorphic to ${\mathbb{R}}^{6g+6+2n}$ the mappinc class groups acts property discontinuous.

Geometry of surface bundles over surfaces

When we bundle over surfaces then the bundle becomes a 4-Mannigfaltigkeit. Question: Given $S_g\to E\to S_h$ where the surfaces have genus $2$ or more. Do such bundle exists which are (from strong to weak):

• Hyperelliptic space?
• Negativ gekrümmte Mannigfaltigkeit?
• Hyperbolic metric space?
• Atoroidal surface bundle

We do a construction of the surface bundle ${\pi }_1(S_h)\subseteq MCG(S_g)$

Theorem

Given a finite cover $S\to X$ annd a branched set $\beta \subseteq X$, there is a finite index subgroup $\Gamma \subseteq MCG(X,\beta )$ (fixing $\beta$) and a lift for the elements of $\Gamma$ to mapping classes of $S$. $l:\Gamma \to Mod(S)$

Theorem

The Birman exact sequence is $0\to {\pi }_1(X)\to MCG(X,\ast )\to MCG(X)\to 0$

This gives us an inclusion of the Point pushes into the mapping class $MCG(X,\ast )$. The two results let us stack “point pushes” (create ann exact sequence with many points)

(Atiyah, Kodaira)

We make a double point push (moving two particles braid like around the surface). This gives an exact sequence $1\to {\pi }_1(Con{g}_2(S_3))\to MCG(S_{3,2})\to MCG(S_3)\to 1$