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1
vote
2answers
288 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
224 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 ...
6
votes
3answers
522 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
285 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
458 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
232 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 ...
18
votes
0answers
434 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
286 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. ...
11
votes
3answers
581 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
398 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
247 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
390 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
260 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
382 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. ...
16
votes
6answers
2k 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
435 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
260 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
271 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
201 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
809 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
148 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
571 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
996 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
591 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
219 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
380 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 ...
2
votes
1answer
296 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
177 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
1k 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 ...
-1
votes
1answer
438 views

Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy 1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $. 2) $ a\bar{b}+c\bar{d}=0 $ There is a ...
13
votes
2answers
324 views

Heegaard splitting of covering hyperbolic manifold.

I am curious about how the Heegaard genus changes after a finite covering. Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that the Heegaard genus of a finite covering of $N$ ...
2
votes
1answer
331 views

Explicit tetrahedralizations of closed 3-manifolds and connections between convex polytopes and hyperbolic knot complements

There are three consistent ways to glue opposite faces of a dodecahedron to get a closed 3-manifold. With a $\frac{1}{10}$ twist we receive the Poincaré dodecahedral space, with a $\frac{3}{10}$ twist ...
0
votes
1answer
424 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
15
votes
3answers
675 views

3-manifolds with solvable fundamental group

Is there a nice reference for the classification of closed 3-manifolds with solvable (nilpotent, abelian, etc.) fundamental group, assuming the Geometrization Conjecture?
10
votes
1answer
906 views

Question about Thurston's paper “A norm for the homology of 3-manifolds”

I have a question about the proof of Theorem 5 in Thurston's paper "A norm for the homology of 3-manifolds". It's the theorem that asserts that the fibered faces are, well, fibered. More precisely, ...
6
votes
0answers
565 views

Expository accounts of the Thurston norm

Other than Thurston's original paper (which I find quite hard to read), are there any expositions of the basic properties of the Thurston norm? In particular, I'm interested in a proof of his ...
4
votes
1answer
191 views

Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...
17
votes
1answer
869 views

Diffeomorphisms vs homeomorphisms of 3-manifolds

For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$, $${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$ a weak homotopy ...
5
votes
1answer
634 views

Dehn surgery on handlebody

Assume $V$ is a handlebody and $C$ be a simple closed curve contained in the interior of $V$. As Sam said, there exists some simple closed curve such that every dehn surgery along it produces a ...
4
votes
1answer
277 views

what is the meaning of “inseparable” in this case

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map. If $i(T^{2})$ is inseparable in $S^{1}\times S^{2}$, then $S^{1}\times S^{2}-i(T^{2})\cong (O\cup K)^{c}$. Here $(O\cup K)^{c}$ is a two ...
3
votes
3answers
439 views

Questions on 3-manifolds with a given boundary

I have the following question: For a given two-dimensional Riemann surface $C$, Is there a way to classify all topologically distinct three-dimensional compact manifolds $M$ whose boundary is $C$, ...
1
vote
1answer
315 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 ...
2
votes
3answers
584 views

what is the meaning of a curve $C$ representing Identity in fundamental group?

Suppose $M$ is a closed 3-manifold, $C\subset M$ is a simple closed curve which represent identity in $\pi_{1}(M)$. Then $C$ bounds an immersed disk in $M$. My question is: When does it bound an ...
9
votes
2answers
596 views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
0
votes
1answer
122 views

is very compact P^2-irreducible 3-manifold homotopy equivalent to a sphere or cell-quotient?

is very compact $P^2$-irreducible 3-manifold homotopy equivalent to a sphere or cell-quotient?
2
votes
1answer
259 views

The Tubular Neighborhood of a Closed Geodesic

Suppose $M_{g}$ is the mapping torus $\Sigma_{g} \times [0, 1]/ (x, 0) \equiv (\tau x, 1)$, where $\Sigma_{g}$ is the hyperbolic space with genus $g,$ and $\tau : \Sigma_{g} \to \Sigma_{g}$ is an ...
6
votes
2answers
286 views

$S^1$-action in three dimensions

Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action. What does this imply for $M$? What are examples except for (products of) spheres?
8
votes
1answer
440 views

Topological rigidity of compact manifolds in dimension three

The Borel Conjecture asserts that homotopy equivalent aspherical closed manifolds are homeomorphic, which is still open in general. But, for three-dimensional manifolds, this conjecture holds (I ...
1
vote
1answer
103 views

Will the rank of fundemantal 3 manifold be decreased is I module the n(n>1) times of a element?

I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible ...