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12
votes
1answer
373 views

Distinguishing 3-manifolds by homologies of covers

In a blog post on ldtopology, a recent arxiv posting of Lins-Lins is discussed. The main argument of that paper is difficult to algorithmically distinguish two 3-manifolds and to that end the authors ...
8
votes
1answer
203 views

Isotopy classes on the disk and mapping tori

Is the following true? "The conjugacy classes of two homeomorphisms of the n-times punctured disk have isotopic representatives iff the associated mapping tori are homeomorphic." By conjugacy class ...
1
vote
0answers
110 views

Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...
6
votes
2answers
377 views

Do different Dehn fillings produce homeomorphic 3-manifolds ?

Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold. Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a hyperbolic structure. ...
6
votes
1answer
317 views

When are isometry groups of hyperbolic 3-manifolds finite?

If $M$ is a finite volume hyperbolic 3-manifold, then its isometry group is finite. I believe this is also true for geometrically finite 3-manifolds. What is the most general condition on a hyperbolic ...
6
votes
0answers
310 views

A conjecture of Thurston and possibly Weeks too

What is the status of the following conjecture: "... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
2
votes
1answer
163 views

Once punctured torus bundles in snappy/twister

I have been trying to learn about snappy's method for encoding once-punctured torus bundles (http://www.math.uic.edu/t3m/SnapPy/manifold.html#snappy.Manifold). As you can see from the link, they are ...
15
votes
1answer
563 views

Can the SL_2 character variety of a three-manifold be nonreduced?

Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$: $$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$ ...
6
votes
1answer
381 views

integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...
7
votes
2answers
212 views

Non-tame 3-manifolds covered by the Euclidean space

An open 3-manifold is tame if it is homeomorphic to the interior of a compact manifold. Is there a (known) example of an open 3-manifold that is not tame, has finitely generated fundamental group and ...
19
votes
2answers
276 views

Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds

Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. ...
6
votes
1answer
368 views

Hyperbolic 3-manifolds with no geometrically finite structure

Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary? I think the answer ...
3
votes
2answers
292 views

The knot whose complement is the Hantzsche-Wendt manifold

Can someone identify the knot whose complement in $\mathbb{S}^{3}$ will produce the Hantzsche-Wendt manifold? Thanks
6
votes
2answers
326 views

Hyperbolic Brunnian links and rectangular cusp shapes

My question is as follows. Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components? Here is what I mean: The Borromean rings form a famous link $B$ (a smooth ...
0
votes
0answers
76 views

projective triangulation and holonomy map

Suppose we have a 3-manifold which is triangulated .what can we deduce if we know that the holonomy map is the identity map around each vertex of the triangulation? (Here we imagine the covering space ...
5
votes
1answer
233 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 ...
0
votes
1answer
235 views

slam-dunk operation

I study surgery theory on 3-manifolds using the text book written by Gompf and Stipsicz.I can't understand \bf{slam-dunk} operation. Let $K_{1}$ be the meridian of a knot $K_{2}$ in $S^3$, and $T$ is ...
2
votes
1answer
175 views

the carrier graph and Heegaard surface

Let $M$ be orientable 3-manifold admitting a Heegaard splitting $V\cup_{S}W$. Let $X$ be a carrier graph of $M$ such that rank($X$)=rank($\pi_{1} M$). Note: A connected graph is called a carrier ...
8
votes
1answer
315 views

Geodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?

I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out. We have ...
4
votes
1answer
240 views

Equivalent to Oriented knot complement conjecture

I would like to see why the following two statements in Kirby's list of problem are equivalent: Statement 1: If $K_1$ and $K_2$ are knots in a closed oriented 3-manifold $M$ whose complements are ...
1
vote
2answers
293 views

faraway curves in surface

Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite. E.Klarreich (see also U.Hamenstadt) ...
6
votes
1answer
237 views

Can bilipschitz models of hyperbolic 3-manifolds be made effective?

In their proof of the Ending Lamination Conjecture, Brock, Canary, and Minsky prove existence of bilipschitz models of hyperbolic 3-manifolds (homeomorphic to a surface times $\mathbb{R}$) depending ...
7
votes
3answers
536 views

Judging whether a finitely presented group is a 3-manifold group?

Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
10
votes
1answer
295 views

Examples of 3-manifolds with RFRS fundamental group

I'm wondering if anyone knows how to construct hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in ...
6
votes
2answers
487 views

heegard diagram

It seems like there is an algorithm to find the Heegard diagram of a 3 manifold obtained by surgery on a link. Also someone told me I can find it in the Gompf and Stipciz's book. But I could not find ...
5
votes
1answer
248 views

What is known about a 3-manifold $M$ when its fundamental group is linear?

Suppose we have a 3-manifold $M$ and its respective fundamental group $\pi_1(M)$. An important question about its fundamental group is to ask if it is linear, i.e. they are isomorphic to a subgroup of ...
19
votes
0answers
447 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 ...
4
votes
1answer
292 views

When are incompressible surfaces isotopic into a two-skeleton?

Haken proved that an incompressible surface in a triangulated irreducible 3-manifold is isotopic to a surface which is normal with respect to the triangulation (Theorie der Normalflächen. Acta Math. ...
12
votes
3answers
657 views

Irreducible homology 3-spheres that bound smooth contractible manifolds

Some examples of irreducible homology 3-spheres that bound smooth contractible 4-manifolds are listed in the comment to problem 4.2 in Kirby's problem list, and all of them happen to occur among the ...
3
votes
1answer
426 views

slam-dunk or Kirby move

I have difficulty understanding the slam-dunk. In slam-dunk you have two knots K1 and K2 with framings r (rational) and n (integer) such that K1 is the meridian of K2. Now we perform surgery on K2 ...
5
votes
2answers
254 views

What is a higher genus analogue of the Pontryagin product?

Given a compact oriented aspherical $3$--manifold $M$ with torus boundary $\partial M\simeq T^2$ (e.g. a knot complement), the condition that the images in $\pi_1 M$ of basis $x,y\in \pi_1 T^2$ under ...
7
votes
2answers
423 views

Isotopy in 3-manifolds

Assume $\Sigma_1$ and $\Sigma_2$ are two embedded compact surfaces (say orientable) in an orientable 3-manifold $M$. Assume $\Sigma_1$ and $\Sigma_2$ are homotopic in $M$. Then are they isotopic?
6
votes
1answer
269 views

What is known about the distribution of average edge-degrees for 3-manifold triangulations (with the number of 3-simplices less than a fixed constant)?

This is my first question on mathoverflow! It relates to a project I'm undertaking with a student. Work by Tamura (extending results by Luo and Stong) shows that for any closed 3-manifold $M$ and any ...
4
votes
3answers
402 views

Second Homotopy Group of Graph Manifolds

A graph manifold is a closed 3-manifold $M$ that admits a finite collection of disjoint embedded tori $\mathcal{T}$ so that $M \setminus \mathcal{T}$ is a disjoint union of Seifert fibred spaces (i.e. ...
19
votes
7answers
3k views

The Chern-Simons/Wess-Zumino-Witten correspondence

I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship. I guess in the condensed matter ...
21
votes
2answers
2k views

Elevator pitch for the Virtual Fibering Theorem

There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm ...
10
votes
3answers
470 views

Torsion in cuspidal cohomology

Following Lemma 2.7 from Vogtmann's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ ...
6
votes
2answers
268 views

Hyperbolic structures on $S\times\mathbb{R}$

Let $S$ be a closed incompressible surface in a finite volume hyperbolic 3-manifold $M$ without cusps. Let $N$ be the cover of $M$ associated to $\pi_1(S) \subset \pi_1(M)$. The cover $N$ is ...
3
votes
3answers
282 views

When is a three-manifold deck transformation group solvable?

Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a rational homology ...
3
votes
1answer
229 views

Is there a torsion element in the homology cylinder group?

The homology cylinder group is the monoid of homology cylinders modulo homology cobordism. I wonder whether there is any known finite order element in this group.
8
votes
3answers
913 views

Why S^3-K and SL(2,R)/SL(2,Z) are diffeomorphic? Here K is a trefoil in S^3.

I've heard this result from my differential manifold class, and I don't know how to prove it. Does anyone know how to construct such diffeomorphism? Please tell me, thanks a lot. Any comments are ...
1
vote
1answer
162 views

Does the coefficient of the meridian determine the coefficient of the longitude?(on Dehn surgery)

I'm studying Dehn surgery, and it says that the coefficient $(p,q)$ which says how the meridian curve on solid torus is attached will determine the entire resulting manifold. I'm wondering whether the ...
6
votes
2answers
646 views

Morse Function on 3-Torus from Heegaard Splitting

It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus ...
5
votes
5answers
1k views

What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?

My question is as stated in the title: What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ? I am particularly interested to know if ...
10
votes
3answers
627 views

Best known Margulis constants?

A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ ...
4
votes
2answers
233 views

Pseudoanosov mapping torus and length of curves.

Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudo-Anosov map $\phi$ in a surface $S$. Is there any way to estimate the length of the geodesic representing a given curve in the surface ...
9
votes
1answer
399 views

Complexity of surfaces bounding knots in 4-ball and 3-sphere respectively

I'm interested in a complexity question related to problems like the slice-ribbon problem. To be specific, if $K \subset S^3$ is a knot, it might be non-trivial yet still bound a smoothly-embedded ...
4
votes
3answers
430 views

teichmuller geodesics and hyperbolic mapping torus

Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a ...
4
votes
1answer
183 views

Rank of a group generated by side-pairing isometries of a polyhedron

Let $P$ be a compact convex polyhedron in $\mathbb{H}^3$. Let $G$ be a group generated by side-pairing isometries of $P$. Is there an algorithm to find the rank of $G$?
23
votes
0answers
2k views

Are there lots of integer homology three-spheres?

The problem of counting combinatorial three-spheres with $N$ simplices has implications for some partition functions in physics (see a paper by Benedetti and Ziegler for more background and ...