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. ...
- 64.2k
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 ...
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 ...
- 34k
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 ...
- 41.2k
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 ...
- 41.2k
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 ...
- 18.7k
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 ...
- 64.2k
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 ...
- 17.2k
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 ...
- 861
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 (...
- 9,554
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 ...
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 ...
- 349
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 ...
- 23.3k
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 ...
- 94.7k
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\...
- 64.2k
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&...
- 22.8k
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)...
- 9,554
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}(...
- 64.2k
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 $S^3$. ...
- 33.1k
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:
- 9,554
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: $(\...
- 12.6k
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 ...
- 3,299
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 ...
- 64.2k
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)....
- 8,509
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 ...
- 6,528
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, ...
- 18.1k
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 $...
- 9,672
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 ...
- 41.9k
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 ...
- 41.2k
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 ...
- 64.2k
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