12
votes
Accepted
Example of three dimensional atoroidal Poincaré duality group with some pathology
One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.
An earlier example is (say) the $(1, 2)$-Dehn filling of the ...
- 26.4k
12
votes
Accepted
Problem 3.14 from Kirby's list
This problem is answered in the literature, with a caveat.
As indicated in the comments, it follows from the orbifold theorem + geometrization conjecture (to handle the case of orbifolds without fixed ...
- 66.8k
11
votes
Accepted
If the universal cover has three boundary components, does it have infinitely many?
Let $M$ be a compact connected 3-manifold with nonempty boundary such that $\pi_1(M)$ is infinite and the universal cover $\tilde{M}$ has finitely many boundary components. I will prove that $\tilde{...
- 43.6k
8
votes
Accepted
Properly embedded surfaces in handlebodies are compressible or boundary compressible?
Suppose that the surface $F$ is properly embedded in the handlebody $V$. We induct on the genus of $V$.
In the base case $V$ has genus zero and so is a three-ball. In this case the desired result ...
- 26.4k
7
votes
Three-dimensional triangulations with fixed number of vertices
Here are some observations that might be useful: not an answer, but too long for a comment.
The Euler characteristic of $S^3$ is zero (this holds for any compact three-manifold without boundary). So ...
- 26.4k
7
votes
Accepted
Branched coverings of non-orientable 3-manifolds
A branched covering must induce a surjection on rational homology, which one can see, for example, by the existence of a transfer homomorphism. So there is no branched covering $S^3 \to \mathbb{R}P^2 \...
- 3,226
7
votes
0-surgery on a fibered hyperbolic ribbon knot
I had a quick look at Maggie Miller's thesis. It appears that the pretzel knots $(\pm 2, n, -n)$ where $n$ is odd are fibered ribbon knots (and they are also hyperbolic). Moreover, it appears that the ...
- 66.8k
7
votes
Accepted
Linking number and intersection number
$\DeclareMathOperator\tX{\widetilde{X}}\DeclareMathOperator\tB{\widetilde{B}}\DeclareMathOperator\tD{\widetilde{D}}\DeclareMathOperator\Z{\mathbb{Z}}$
In fact, $B$ must intersect $D$ at least $|\text{...
- 43.6k
6
votes
Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?
This follows now from Theorem 1.15 of this paper for $n>1$. To clarify, Theorem 1.15 is stated for framed instanton homology $I^\#(S^3_{1/n}(K))$. If instanton Floer homology $I(S^3_{1/n}(K))=0$, ...
- 66.8k
6
votes
Accepted
Slice knots in 3-manifolds
Suppose you know that the universal cover of $Y$ embeds in $S^3$, i.e. is $S^3-A$ for some $A$. For example, this happens when $Y$ is a connected sum of lens spaces. (I'm thinking this is always true ...
- 19.2k
6
votes
Accepted
Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
This fact doesn’t need $M$ to be hyperbolic. It just needs one general theorem about 3-manifold topology, namely the Scott core theorem.
Let $N\to M$ be the covering space corresponding to the ...
- 24.2k
5
votes
Accepted
Residual finiteness and a gluing problem
Thurston never finished his project, hence, we cannot know for sure what exactly did he have in mind in this part of the diagram. Here is what we know:
Fundamental groups of good compact 3-...
- 9,748
5
votes
Euler number of a Seifert bundle as a generalization of an Euler number of a circle bundle over a surface
Why is the Euler number of a Seifert bundle a "natural" generalization of a circle bundle over a surface?
I can see at least three reasons:
A circle bundle is a Seifert manifold with no ...
- 10.5k
5
votes
Accepted
Volume of the Weeks manifold and of the 5.2 knot complement
One explanation is to look at the "spun triangulation" SnapPy gives to the Weeks manifold. This is composed of two ideal tetrahedra with volumes ~0.83828404504 and ~0.10442331774 adding to ...
- 26.4k
5
votes
Proof of Giroux's correspondence
I haven't read it carefully, but the new paper here by Licata-Vértesi appears to be (finally) a complete proof, albeit one that is different from the original.
- 43.6k
4
votes
Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?
In fact, if $C = \text{rank}(I_*(S^3_0(K)))$, we have $$f(k) = \text{rank}(I_*(S^3_{1/k}(K))) = kC.$$ This holds with any coefficient field.
Floer's exact sequence gives us an exact triangle relating $...
- 9,388
4
votes
Elevator pitch for the Virtual Fibering Theorem
The Virtual Fibring theorem provides the topological classification of closed 3-manifolds.
Very roughly, after passing to finite covers, we can build 3-manifolds by gluing together simpler pieces that ...
- 24.2k
4
votes
Betti numbers of non-orientable $3$-manifolds
In the nonorientable case the “half lives, half dies principle” has the following form:
$$\dim \ker \left( H_{1}(\partial M;\mathbb{Q}) \to H_{1}(M;\mathbb{Q}) \right) + \dim \ker \left( H_{1}(\...
- 71
4
votes
Accepted
Non compact Seifert manifolds
This is copied, in part, from the comments. So I've made it community wiki. Please feel free to edit and improve.
The consensus in the comments is that "the classification in the non-compact ...
4
votes
Accepted
Guts of 3-manifolds for sutured manifolds and pared manifolds
Edited: to reflect the correct definitions.
Question 1: Why are the guts well-defined?
Answer 1: By the JSJ theory there is a unique collection of $I$-bundles (and Seifert fibered spaces) that ...
- 26.4k
3
votes
Rigidity/flexibility of Sol-structures on closed 3-manifolds
Let's look at the case of Solv manifolds $M$ which are torus bundles over the circle. (All Solv manifolds are finitely covered by such bundles.) The square of the length element is as follows:
$$ds^...
- 26.4k
3
votes
Accepted
Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds
$\newcommand{\PSLt}{\widetilde{\mathrm{PSL}_2}}$Manifolds with $\PSLt$ geometry can have non-trivial moduli. For example, suppose that $S$ is a closed, connected, oriented surface with genus at least ...
- 26.4k
3
votes
"canonical" framing of 3-manifolds
The answer is a paper aptly titled Canonical framings for 3-manifolds by Rob Kirby and Paul Melvin.
Here is a 2012 email correspondence between me and Kirby concerning that paper (glad I didn't walk ...
- 17.2k
3
votes
"canonical" framing of 3-manifolds
This is maybe not the answer you're looking for, but it's certainly too long for a comment.
First, there's no contradiction here, and nothing too strange. A 2-framing on $M$ is a framing of $TM\oplus ...
- 10.5k
3
votes
When are homologous embedded surfaces in 3-manifolds related by embedded cobordisms?
This “sort of thing” is proven by Kakimizu [1992, 2005] and by others in many later papers. If you want more references then Google “The Kakimizu complex is connected” (or simply-connected, or ...
- 26.4k
3
votes
Three-dimensional triangulations with fixed number of vertices
This is going to be a long one. I believe I now have a triangulation of the $3$-sphere with $28$ tetrahedra, $56$ triangles, $32$ edges and $4$ vertices, obeying all of the rules.
Our triangulation ...
- 151k
3
votes
Mappings of reducible 3 manifolds with boundary
Some care is required in phrasing the question (and the desired form of the answer). For consider the following two examples.
Suppose that $M = V_g$ is the handlebody of genus $g$. For example, $M$ ...
- 26.4k
3
votes
Counterexample to mostow rigidity theorem
Yes, I believe there are many.
For example, if you think of hyperbolic $2$-space as a geodesic subspace of hyperbolic $3$-space, any group of hyperbolic isometries of hyperbolic $2$-space extends ...
- 43.1k
2
votes
The works of González-Acuña and Duchon from 70s and 80s
All of these papers are currently available online.
González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79.
González-Acuña, F. On homology spheres. Thesis (Ph.D.)...
- 29.8k
2
votes
Realizable geometrically finite hyperbolic 3-manifolds with prescribed conformal boundaries
The answer to question one and question two are both "yes". Here is a very special case:
Suppose that $M$ is a connected oriented compact three-manifold with boundary. Suppose further that ...
- 26.4k
Only top scored, non community-wiki answers of a minimum length are eligible