All Questions
7 questions
13
votes
1
answer
2k
views
Incompressible surfaces in an open subset of R^3
Let $U$ be a connected open subset of $\mathbb R^3$. Furthermore, we have:
$\mathbb R^3\setminus U$ has exactly two connected components (thus by Alexander duality, $H_2(U;\mathbb Z)=\mathbb Z$).
$U$...
- 18.7k
13
votes
1
answer
514
views
Geometric intersection with incompressible surfaces
Let $M$ be a oriented compact $3$-manifold, closed or with boundary.
For any incompressible surface $F$, define a function $i_F$ on the set of homotopy classes of closed curves in $M$ by $$i_F (\alpha)...
- 492
10
votes
4
answers
1k
views
Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ ?
Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ where $\Sigma$ is any closed surface? I know of the Hatcher-Thurston classification of incompressible surfaces in 2-bridge ...
- 103
4
votes
1
answer
88
views
$\partial$-incompressibility of a surface obtained when attaching a 2-handle to an irreducible 3-manifold produces a reducible 3-manifold
This question arises in my previous question.
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $\alpha\subseteq \partial M$ be a simple closed curve, which is ...
- 605
4
votes
1
answer
145
views
the local structure of an immersed incompressible surface
Assume that $M$ is a closed, irreducible, orientable 3-manifold. Suppose that we have a closed, immersed, incompressible surface $F$ of genus at least 1. Since we only required $F$ to be immersed in $...
- 841
4
votes
1
answer
217
views
Constructing a "nice" cobordism
Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties:
1) $M_g$ is an ...
- 1,255
2
votes
0
answers
124
views
Non Seifert incompressible surfaces detected by ideal points
Given a 3-manifold with toric boundary, the Culler-Shalen theory associates an incompressible surface to any ideal point of its character variety. From the proof of the Neuwirth conjecture, one knows ...
- 223