All Questions
12
questions
5
votes
1
answer
374
views
Linking number and intersection number
Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
- 125
6
votes
2
answers
365
views
"canonical" framing of 3-manifolds
In Witten's 1989 QFT and Jones polynomial paper, he said
Although the tangent bundle of a three manifold can be trivialized, there is no canonical way to do this.
So if I understand correctly, ...
- 447
1
vote
1
answer
238
views
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory
As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
- 131
3
votes
1
answer
221
views
Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...
- 1,097
7
votes
1
answer
384
views
Implications of Geometrization conjecture for fundamental group
Hempel proved that Haken manifolds have residually finite fundamental groups. With the Geometrization conjecture, this now holds for any compact and orientable 3-manifold.
How exactly does the ...
- 121
10
votes
2
answers
2k
views
Parallelizability of 3-manifolds
Robert Bryant's answer here ( https://mathoverflow.net/a/149496/85500 ) states that any orientable 3-manifold is parallelizable.
Previously I was under the impression that only closed (compact & ...
- 2,712
5
votes
1
answer
204
views
Group of parallelizations of $M^3$ finitely generated?
Let $M^3$ be a compact orientable 3-manifold. Then $TM$ is trivial and let's go ahead and fix a trivialization $\tau : M \times \mathbb{R}^3 \to TM$. Then given a map $g : (M, \partial M) \to (SO(3),...
- 5,339
1
vote
1
answer
961
views
On compact, orientable 3-manifolds with non-empty boundary
I recall my Professor having stated something along the lines of the following, but I am not quite certain about the precise statement she gave:
Let $M$ be a compact, orientable 3 manifold with non-...
- 1,245
5
votes
2
answers
1k
views
Gluing two 3 manifolds along their boundary
Let $X,Y$ be two compact, smooth, orientable 3 manifolds, each with an incompressible boundary component diffeomorphic to some genus $g $ surface $S_g$. Under an orientation-reversig diffeomorphism $...
- 1,245
26
votes
6
answers
3k
views
How to get convinced that there are a lot of 3-manifolds?
My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...
- 2,636
8
votes
1
answer
758
views
Seifert surfaces via Alexander duality
If we take a knot $K$ in $S^3$, there are several ways to construct the associated Seifert surface. One way, which I am not familiar with, I just came across in a paper I am reading. It goes like ...
- 4,375
4
votes
3
answers
830
views
Is a compact, connected, orientable 3-manifold with $\mathbb{Z}^K$ fundemental group uniquely determined?
According to the Kneser-Milnor prime decomposition theorem for 3-manifolds, any compact, connected, orientable 3-manifold $M$ is diffeomorphic to $S^3 / \Gamma_1$ # $\cdots$ # $S^3/ \Gamma_n$ # $(S^2 \...
