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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. ...
• 61.3k

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 ...
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 ...
• 33.3k

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 ...
• 39.8k
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 ...
• 17.3k

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 ...
• 61.3k
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 $\mathbb{C}$...
• 98.1k
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 ...
• 17k
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 (...
• 8,774

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 ...

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 ...
• 831
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 ...
• 93.8k
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• 12.4k
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 ...
• 3,172
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:
• 8,774
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)....
• 7,421
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 ...
• 61.3k
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 ...
• 5,794