# Questions tagged [seifert-fiber-spaces]

The seifert-fiber-spaces tag has no usage guidance.

20
questions

6
votes

1
answer

146
views

### Euler number of a Seifert bundle as a generalization of an Euler number of a circle bundle over a surface

In classic, Euler numbers associated to circle bundles over a fixed surface classify all possible such bundles. But the construction of Euler class in general requires the fact that any fiber bundle ...

4
votes

1
answer

202
views

### Non compact Seifert manifolds

A Seifert manifold $M$ is a $3$-dimensional orientable smooth manifold with an effective circle action with no fixed points.
Closed connected Seifert manifolds are classified up to an equivariant ...

15
votes

4
answers

1k
views

### Cohomology ring of mapping torus

A mapping torus, $M \rtimes_\varphi
S^1$, is a fiber bundle over $S^1$ with fiber $M$, where $\varphi$ is an element of mapping class group of $M$, describing the twist around $S^1$.
For $M=S^1\times ...

3
votes

1
answer

172
views

### How to find the JSJ decomposition in the plumbing tree model of a graph manifold?

A graph manifold can be obtained by plumbing circle bundle over surfaces, where the number in the plumbing tree denotes the Euler number of the bundle (see the picture for an example). The boundary of ...

4
votes

1
answer

193
views

### Mapping class groups of Haken Seifert 3-manifolds (not small)

This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $M$, in particular how to systematically compute it.
I want to ...

4
votes

1
answer

405
views

### Reference sought for Seifert fiber spaces

I seek a reference for what is surely a well known basic result about Seifert fibered 3-manifolds. Namely they are all obtained by Dehn-surgery along a regular Seifert fiber (and the surgery slope is ...

4
votes

1
answer

195
views

### Seifert fiber space with homotopically trivial generic fiber

Let $X$ be a Seifert fiber space, that is, a 3-manifold which is a circle bundle over a 2-orbifold. Suppose all generic fiber of $X$ is homotopically trivial, can we prove that the universal cover of $...

5
votes

2
answers

403
views

### Zero surgery on a Seifert fiber space

I have a problem with understanding what is a neighbourhood of a singular fiber in a Seifert fibered space coming from the zero surgery. For me a 3-manifold $Y$ is a SFS if it has a decomposition into ...

9
votes

1
answer

1k
views

### On the geometrization of double branched covers

I recently got into Lickorish's paper Prime knots and tangles and a question, which I didn't have the first time I read it, naturally emerged.
The Thurston-Perelman Geometrization Theorem asserts ...

3
votes

2
answers

549
views

### Orbifold of the three-sphere (and lens spaces)

Think of the three-sphere as given by $\lbrace|z|^2+|w|^2=1, \;z,w\in \mathbb{C}^2\rbrace$. We can regard it in terms of Hopf coordinates
\begin{align*}
z&= \cos(\theta/2)e^{i(\phi+\psi)}\\
w&=...

10
votes

3
answers

622
views

### Spin 4-manifold bounded by a mapping torus of tori

Consider a smooth torus endowed with the non-bounding spin structure. Pick a basis of its first homology and a diffeomorphism inducing the S-transformation
$\left(\begin{array}{cc} 0 & 1 \\-1 &...

5
votes

0
answers

100
views

### Reference Request: Mapping Class Group of Seifert-Fibered spaces

It seems to be a well understood and old topic, but even after a few days of searching, I am having trouble finding a good/more pedagogical introduction to Mapping Class Group of Seifert-Fibered ...

6
votes

4
answers

568
views

### Structure of foliations of codimension 2 on three dimensional torus

Is it possible to have a one-dimensional foliation on three dimensional torus such that the foliation has a trefoil knot as its leaf?
Moreover, does a one dimensional foliation on three dimensional ...

2
votes

0
answers

457
views

### Trefoil Knot Seifert Minimal Surface Equation

I am not very familiar with knot theory nor with minimal surfaces, so I already apologize if my question appears too naive or simple :). I am trying to do the following:
Starting from a real ...

6
votes

2
answers

398
views

### Is it known which links have Seifert fibered complements?

I believe many such links can be constructed by looking at a foliation similar to the hopf fibration, but the wrapping leaves replaced with $(p,q)$ torus knots. However, I'm interested in particular ...

4
votes

1
answer

587
views

### Seifert fiberable manifolds with several Seifert fiberings

I have a question on Theorem 2.3 on page 34 of Hatcher's notes on 3-manifolds:
Hatcher: Notes on Basic 3-Manifold Topology.
Regarding the class d), it follows from Proposition 2.1 on page 31, that $M(...

13
votes

1
answer

1k
views

### Mapping class groups of small Seifert-fibred 3-manifolds

Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location?
For most Seifert manifolds working out the mapping class group is easy-enough (...

3
votes

1
answer

323
views

### Classification of fiber-preserving branched covers between Seifert fibered integer homology spheres

Is there an easy classification (and proof) of the possible branched covers between Seifert fibered integer homology spheres which are fiber-preserving and branched over fibers (or at least what the ...

8
votes

2
answers

570
views

### Seifert Fibrations and their associated Spectral Sequence

In a somewhat limited setting, a Seifert Fibre Space is a 3-manifold $M$ with a "nice" decomposition into circles (http://en.wikipedia.org/wiki/Seifert_fiber_space). That is, $M$ is decomposed into ...

5
votes

2
answers

569
views

### Getting rid of exceptional fibers by passing to finite covers?

Consider a Seifert fiber space. Is it always possible to find a finite cover that is a circle bundle and the preimage of any fiber is a finite union of circles?