53
votes

Accepted

### Thurston's 24 questions: All settled?

A nice summary of the status of these problems may be found here:
Otal, Jean-Pierre, William P. Thurston: ``Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", Jahresber. Dtsch. ...

31
votes

### Shing-Tung Yau's doubts about Perelman's proof

First, let me make some preliminary remarks. We sometimes like to think that "being proved" is a black-and-white property, but in fact there are shades of gray. At one extreme are things ...

Community wiki

27
votes

Accepted

### Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?

Quoting Theorem F of this paper by Ulrich Koschorke:
For any self-transverse immersion $j$ of a closed 3-manifold $M$ into $\mathbb{R}^4$ the following integers are equal modulo 2:
the ...

25
votes

### Homotopy type of Diff(ℝP³)

This was answered by Bamler and Kleiner, who proved more generally that the diffeomorphism group of any spherical space form deformation retracts to its isometry group. This in particular gives a new ...

24
votes

### Thurston's 24 questions: All settled?

They have all been resolved in some fashion with the exception of problem 23, which asserts that the volumes of all hyperbolic 3-manifolds are not rationally related. We know basically nothing about ...

22
votes

Accepted

### $S^3$ as cyclic branched cover of itself

The statement that for arbitrary K in $S^3$, if for some $n \ge 2$, the n-fold cyclic branched cover is $S^3$ (or in some versions, a homotopy 3-sphere) then K is the unknot, was known as the Smith ...

22
votes

### Examples of interesting non orientable closed 3-manifolds

One gets non-orientable closed 3-manifolds by taking a non-orientable surface, and crossing with $S^1$, such as $P^2\times S^1$.
In fact, the geometrization theorem hasn't been proven completely for ...

21
votes

Accepted

### What should I cite for the Poincaré conjecture?

I think it is customary to cite at least the first two papers ("The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds"). See this for ...

20
votes

### What are the implications of the simple loop conjecture?

The simple loop conjecture can be viewed as a statement about how to construct all surfaces in a 3-manifold.
Fix any orientable 3-manifold M. There are two well known constructions that produce ...

20
votes

Accepted

### Example of homeomorphism of $3$-manifolds

Here is a sequence of moves that gets you from left to right. (Pictures are done with Frenk Swenton's Kirby calculator.)
First, we convert from rational to integral surgery.
Then we blow up twice (...

20
votes

### Shing-Tung Yau's doubts about Perelman's proof

The question is, unavoidably, somewhat ambiguous. Here are seven points, hopefully rather objective:
There are well-known expositions of Perelman's work by Cao & Zhu, Kleiner & Lott, and ...

Community wiki

19
votes

### Drawing of the eight Thurston geometries?

We have recently started working on visualizing Sol.
Sol is defined by the following metric in $\mathbb{R}^3$: $ds^2 = (e^zdx)^2 + (e^{-z}dy)^2 + dz^2$
I think it is quite easy to see what is going ...

19
votes

Accepted

### What are the implications of the simple loop conjecture?

I would motivate the simple loop conjecture as follows. (I'm fairly idiosyncratic about this; I fear I'm going to turn off many 3-manifold topologists.)
As well as understanding spaces, we want to ...

19
votes

Accepted

### Random links and $3$-manifolds

There is no universally accepted model of random three-manifolds (or random knots/links) for that matter, however, hyperbolicity is pervasive in all known models. The most popular (but not really ...

19
votes

Accepted

### Hyperbolic $3$-manifold groups that embed in compact Lie groups

All closed hyperbolic 3-manifold groups embed into a compact Lie group.
To prove this, note first of all that given a hyperbolic 3-manifold $M$, it suffices to show that a finite-index subgroup $G\...

19
votes

### 3-manifold with fundamental group $\mathbb Z$

No. For example, take a copy of $S^1 \times S^2$ and remove the interior of a closed, nicely embedded, three-ball.
You will need to add the hypothesis of irreducibility (to rule out "punctures&...

18
votes

Accepted

### Morse number of the Poincaré homology sphere

It's 6: you need one critical point of index 0 and index 3, and two of index 1 and index 2.
It's at least 6, since the only 3-manifolds with lower Morse number are the 3-sphere (2) and lens spaces (4)...

18
votes

Accepted

### On trivial mapping class group of 3-manifolds

Dave Gabai proved that the mapping class group of a closed hyperbolic 3-manifold is isomorphic to its isometry group. For a hyperbolic knot $K$ without any symmetries, for large enough $n$, $S^3_{1/n}(...

17
votes

### How to get convinced that there are a lot of 3-manifolds?

Here are two examples suggesting the complexity of the world of $3$-manifolds.
The first is the classical result that any $3$-manifold can be obtained by integral surgery on a link in $S^3$. ...

17
votes

Accepted

### Simple question on Kirby move

Yes, there is a simple way. Below is a sequence of pictures illustrating the procedure (created using Kirby calculator).
$5_2$:
Blowup at the clasp:
Isotopy:
Blowdown the purple unknot:

17
votes

Accepted

### How many homotopy types of lens spaces L(p,q) if the given integer p is not prime?

No; in fact, there can be arbitrarily many homotopy types.
The theorem you quote says that the number of homotopy types, for a given $p$, is the same as the size of the following quotient group: $(\...

17
votes

Accepted

### Classification of closed 3-manifolds with finite first homology group?

The answer is no by Yves' comments. Let me add that there are plenty of explicit constructions of closed hyperbolic 3--manifolds with finite homology, and this is a generic phenomenon (for example ...

16
votes

Accepted

### Simple proof for property R conjecture

Property R was reproved by Gordon and Luecke in the course of solving the knot complement problem - see Corollary 3.2. They prove the stronger result (as did Gabai) that zero-frame surgery on a knot ...

16
votes

Accepted

### Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?

Here is an argument that "most" points in the $SL(2, {\mathbb C})$-character variety $X(F)$ of the surface $F$ do not correspond to representations extendible to 3-manifold groups (as in the question)....

15
votes

Accepted

### Searching for a Thurston paper with egg / 3-manifold analogy?

I believe you're looking for page 211 of Hyperbolic Structures on 3-Manifolds I: Deformation of Acylindrical Manifolds (Annals of Math., 1986), which includes the following paragraph:
A complete ...

14
votes

### Drawing of the eight Thurston geometries?

Not only could you see the eight geometries at ihp's exposition esthetopies last summer, but you could also hear them.
The exposition is now over but the pictures are on the site of Pierre Berger, ...

14
votes

### How to get convinced that there are a lot of 3-manifolds?

Maybe by looking first at homology $3$-spheres and in particular to Brieskorn manifolds $M(p,q,r)$:
the link of the singular point $(0,0,0)$ of the hypersurface
$$z^p_1+z_2^q+z^r_3=0$$
with integers $...

14
votes

Accepted

### Classification of knots by geometrization theorem

You have all the tools to compute the geometric decomposition of knot and link exteriors in the software Regina. I'm one of the authors, although my hands haven't been over that part of the code very ...

14
votes

Accepted

### Reference request for wild 3-manifolds

The only two books that I know of that focus on wild/pathological aspects of 3-manifolds are Bing's "Geometric topology of 3-manifolds" and Moise's "Geometric topology in dimension 2 and 3". But if ...

14
votes

Accepted

### Two Dehn fillings yielding the same lens space?

This is a case of the oriented knot complement problem in lens spaces, also called the cosmetic surgery problem. If $M(r)\cong M(r')$ preserving orientation, then these form a purely cosmetic (or ...

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