All Questions
33 questions
19
votes
0
answers
575
views
The oriented homeomorphism problem for Haken 3-manifolds
Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it'...
- 25k
18
votes
3
answers
1k
views
Classification of knots by geometrization theorem
I read this interview with Ian Agol, where he says:
"...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots."
My question is:
How does this ...
- 1,314
14
votes
1
answer
937
views
Classification of 3-dimensional manifolds with boundary
It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as
$$\mathcal{M}=P_{1}\#\dots\# P_{n}$$
where $P_{i}$ are prime manifolds, i.e. ...
- 1,429
11
votes
3
answers
821
views
Random links and $3$-manifolds
In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
A non-trivial connected sum $M_1\# M_2$ ...
- 1,314
11
votes
1
answer
749
views
Searching for a Thurston paper with egg / 3-manifold analogy?
I remember coming across some article of Bill Thurston’s where he describes a 3-manifold (with boundary?) as being like an egg. In my recollection the interior of the egg, the shell, and even the ...
- 1,996
11
votes
1
answer
697
views
Did Milnor and Thurston write anything else about characteristic numbers for 3-manifolds?
In Characteristic numbers for 3-manifolds Milnor and Thurston define a characteristic number and this is cited in ch. 6 of Thurston's notes when discussing the Gromov approach to Mostow rigidity.
The ...
10
votes
3
answers
3k
views
References about 3-manifolds
I am working on a subject of geometric group theory closely related to 3-manifolds, and in order to understand these links, I am seeking a good reference about 3-manifolds, as self-contained as ...
- 2,371
10
votes
1
answer
881
views
3-manifolds with isomorphic fundamental groups
There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...
- 5,259
9
votes
3
answers
797
views
Reference request for wild 3-manifolds
I’m looking for a text on 3-manifolds that focuses on wild/pathological objects, similar to Bing’s work in the field. I know basic algebraic topology (homotopy, homology, cohomology) and have read ...
- 2,069
9
votes
1
answer
637
views
Reference request: A knot is tame if and only if it has a tubular neighbourhood
Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...
- 93
9
votes
1
answer
296
views
The works of González-Acuña and Duchon from 70s and 80s
I would like to access the following two works of González-Acuña from 1970:
González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79.
and
González-Acuña, F. On ...
- 1,292
8
votes
1
answer
281
views
Non-compact three-manifolds with the same proper homotopy type are homeomorphic?
I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
- 1,097
7
votes
2
answers
366
views
Boundary of a $4$-manifold and the fundamental group
I am trying to learn $4$-manifolds with boundaries and I don't know much about this topic so these questions may be silly. Given a $4$-manifold $M$ with a boundary say $N$,
Assume $\pi_1(N)$ is known,...
- 1,177
7
votes
1
answer
636
views
Knots: locally flat, PL and smooth
In the classical dimension (knots in $S^3$), it is considered standard (I think?) that the following sets are in bijective correspondence:
locally flat knots up to ambient isotopy;
PL-knots up to PL ...
- 1,033
7
votes
0
answers
156
views
Two papers on surface diffeomorphisms
The following two papers appeared in the reference of a paper i was reading.It seems that neither is published formally.Is there a website where i could find them?
A. Casson, Cobordism Invariants of ...
- 103
6
votes
2
answers
912
views
"Well-known fact" that every irreducible 3-manifold with non-empty boundary has an incompressible surface
I have seen in several sources that this results holds, however none of them included the proof. Does anyone know where I can find one?
Also, it would be great if someone could provide me with a ...
- 214
6
votes
2
answers
395
views
Dual surfaces of a first cohomology class of a 3-manifold
Let $M$ be closed 3-manifold and $\alpha\in H^1(M;\mathbb Z_2)$ an arbitrary element. (In my case we know that $M$ is non-orientable and $\alpha^3=0$.) It is well known that there is a closed 2-...
- 1,095
6
votes
1
answer
800
views
Geometrization & JSJ decomposition with boundary
Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...
- 125
5
votes
1
answer
188
views
Properly embedded surfaces in handlebodies are compressible or boundary compressible?
I've read in a couple of different places (a paper and a blog) the following fact:
if $F$ is a surface, properly embedded in a three-dimensional handlebody of genus at least two, then $F$ is either ...
- 421
5
votes
1
answer
439
views
Rotation part of short geodesics in hyperbolic mapping tori
Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
- 1,601
5
votes
1
answer
283
views
Homology of spherical $3$-manifold group
I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true.
Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
- 179
5
votes
2
answers
406
views
Unknotting tunnels in surface bundles
Given a once-punctured surface $F$ and an orientation preserving self homeomorphism $\phi$, let $M_\phi$ be the bundle over $S^1$ with fiber $F$ and monodromy $\phi$.
In Sakuma's survey article The ...
- 1,601
4
votes
1
answer
119
views
Enumeration of three dimensional spherical good orbifolds covered by Nil, sol and E3
Is there in the literature a list of three dimensional spherical, good orbifolds covered by nil, Sol and E3, and their algebraic topological invariants? (Homology, orbifold fundamental group).
- 2,630
3
votes
2
answers
950
views
hyperbolic 3-manifold of finite volume
Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
...
- 1,629
3
votes
1
answer
516
views
Ambiguity in the unoriented knot connected sum
It is well known that there might be ambiguity in the unoriented knot connected sum if the knots concerned are not invertible.
E.g., consider 8_17, the only knot with crossing number 8 which is non-...
3
votes
1
answer
235
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
3
votes
1
answer
152
views
Reference request: Stallings-Epstein-Waldhausen construction
I am looking for a reference for the Stallings-Epstein-Waldhausen construction (constructing an incompressible surface in a 3-manifold from a nontrivial splitting of the fundamental group).
I know of ...
- 125
2
votes
1
answer
709
views
Smooth structures on closed $3$-manifolds are unique up to diffeomorphism?
Hi!
I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the ...
- 23
1
vote
2
answers
350
views
Commutativity in the Fundamental Group and Knot Theory
Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...
- 1,108
1
vote
2
answers
335
views
The moduli space of finite volume hyperbolic 3-manifolds?
By finite volume hyperbolic 3-manifold, I do mean $M=\mathbb{H}^{3}/\Gamma$ where $\Gamma$ is a torsion-free Kleinian group such that the hyperbolic volume $Vol(M)<\infty$.
I will call
$$\mathcal{M}...
- 223
1
vote
1
answer
124
views
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
My question is in the tittle:
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
If the answer is yes, is there a reference for this.
- 11
1
vote
1
answer
256
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
1
vote
1
answer
773
views
Dehn twist generators for mapping class group of a genus zero surface with boundary
Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(S_{0,n})$, the mapping class group of a genus $0$ surface with $n$ boundary components, fixing the ...
- 143