52 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. ...
user avatar
  • 61.3k
29 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 ...
user avatar
  • 33.3k
23 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 ...
user avatar
  • 39.8k
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 ...
user avatar
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 ...
user avatar
  • 61.3k
21 votes
Accepted

Geometry of the space of circles in the Euclidean plane

Things will simplify if you just consider the circles on the Riemann sphere $S^2 = \mathbb{C}\cup\{\infty\}$, for your space is simply the space of circles on the sphere (with the lines in $\mathbb{C}$...
user avatar
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 ...
user avatar
  • 17k
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 (...
user avatar
  • 8,774
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

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 ...
user avatar
  • 831
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 ...
user avatar
  • 93.8k
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\...
user avatar
  • 61.3k
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&...
user avatar
  • 19.1k
18 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 ...
user avatar
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)...
user avatar
  • 8,774
18 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 ...
user avatar
  • 22.3k
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 $S^3$. ...
user avatar
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: $(\...
user avatar
  • 12.4k
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 ...
user avatar
16 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:
user avatar
  • 8,774
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)....
user avatar
  • 7,421
15 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 ...
user avatar
  • 61.3k
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 ...
user avatar
  • 5,794
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, ...
user avatar
  • 17k
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 $...
user avatar
  • 9,482
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 ...
user avatar
  • 40.7k
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 ...
user avatar
  • 39.8k
13 votes
Accepted

How many knots are there with hyperbolic volume less than a given constant

Here is an expansion of Ian's answer. Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in ...
user avatar
  • 19.1k
13 votes
Accepted

Rational homology sphere that is not Seifert manifold

By Thurston, all but finitely many $(p,q)$-surgeries on a hyperbolic knot in $S^3$ result in hyperbolic rational homology spheres for $p\neq 0$. In particular there are infinitely many integral ...
user avatar

Only top scored, non community-wiki answers of a minimum length are eligible