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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. ...
Ian Agol's user avatar
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33 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 ...
26 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 ...
Andy Putman's user avatar
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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 ...
Andy Putman's user avatar
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23 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 ...
Danny Ruberman's user avatar
23 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 ...
Ian Agol's user avatar
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23 votes

$3$-manifold that is a surgery on a knot

This is an extensively studied question and is far from being understood in general. Here are some other conditions beyond the fact that $H_1(M)$ is cyclic. the fundamental group should have weight 1,...
Ian Agol's user avatar
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21 votes
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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 ...
Myshkin's user avatar
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21 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 ...
20 votes
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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\...
Ian Agol's user avatar
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20 votes
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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 (...
Marco Golla's user avatar
  • 10.5k
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 ...
Zeno Rogue's user avatar
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&...
Sam Nead's user avatar
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18 votes
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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}(...
Ian Agol's user avatar
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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:
Marco Golla's user avatar
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17 votes
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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: $(\...
Greg Martin's user avatar
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17 votes
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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 ...
Jean Raimbault's user avatar
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)....
Moishe Kohan's user avatar
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15 votes
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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 ...
Thomas Bloom's user avatar
  • 6,628
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, ...
coudy's user avatar
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14 votes
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Mapping class group of certain 3-manifolds

Since you write ${\rm Diff}_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientation-preserving. Every diffeomorphism of $M$ can be isotoped to take fibers to ...
Allen Hatcher's user avatar
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 ...
Andy Putman's user avatar
  • 43.9k
14 votes
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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 ...
Ian Agol's user avatar
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14 votes
Accepted

Definition of Thurston's skinning map

Let’s simplify to the case where $M$ has exactly one boundary component, say $\partial M = S$. So the hyperbolic structures on $M$ are parametrised by the conformal structures on $S$. Fix one such ...
Sam Nead's user avatar
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13 votes
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Definition of cusped manifold?

Cusped manifolds are noncompact complete hyperbolic manifolds with finite Riemannian volume. More precisely, a cusped hyperbolic n-manifold is a Riemannian manifold (without boundary) of constant ...
Jean Raimbault's user avatar
13 votes

Parallelizability of 3-manifolds

Edit: the following argument is wrong. Please ignore this post. I think that all orientable three-manifolds $M^3$ are parallelizable. Here is a proof in the smooth setting. Assume we know that it ...
Mizar's user avatar
  • 3,106
13 votes
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Cobordism and Kirby calculus

As Golla pointed out that since every smooth $4$-manifold has a handle decomposition, you can draw a Kirby diagram. See the following pretty nice picture from Akbulut's lecture notes (now it is a ...
Oğuz Şavk's user avatar
  • 1,292
13 votes

$0$-surgeries on trefoil and figure-eight

They can also be distinguished geometrically. Both knots are genus one fibered knots, so both $M$ and $N$ are torus bundles over the circle. The complement of the figure eight is hyperbolic, so the ...
Josh Howie's user avatar
  • 1,617
13 votes
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Higher homotopy groups of irreducible 3-manifolds

An irreducible 3-manifold $M$ is aspherical if and only if it's not a finite quotient of $S^3$, which in turn is equivalent to having infinite fundamental group. Essentially you've already outlined ...
Marco Golla's user avatar
  • 10.5k
13 votes
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Where was it first shown that every homotopy self-equivalence of $S^1\times S^2$ is homotopic to a homeomorphism?

My guess is that the oldest reference might be Pontryagin's 1941 paper on the homotopy classification of maps from a 3-dimensional complex to the 2-sphere, the English version of which is in Recueil ...
Allen Hatcher's user avatar

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