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Results tagged with gt.geometric-topology
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user 1650
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
2
votes
Accepted
When is a 2-bridge knot hyperbolic?
All two-bridge knots are hyperbolic except for the $(2, k)$ torus links. For an “as simple as possible” (but still fairly difficult) proof, see Theorem 10.17 of Purcell’s book Hyperbolic Knot Theory.
- 28.2k
19
votes
Is there a continuous partition of space into circles?
As a partial answer, there is no smooth partition of $\mathbb{R}^3$ into smooth loops. For, consider the quotient space; it is necessarily a two-orbifold (locally a surface, possibly having isolated …
- 28.2k
3
votes
Accepted
Existence of orientable finite volume complete cusp hyperbolic 3-manifolds $\mathbb{H}^3 / \...
Yes, all hyperbolic three-manifold fundamental groups can be generated by loxodromic elements. For, suppose that $\Gamma = \{ \gamma_i \}$ is a generating set. Take $\gamma$, a loxodromic element wh …
- 12.3k
12
votes
Accepted
Can I endow the following 3-manifold with a hyperbolic metric?
This three-manifold can also be constructed by taking a genus two surface $S$, crossing with the interval $I$ to get $S \times I$, and attaching a pair of one-handles both of which connect $S \times \ …
- 28.2k
6
votes
Accepted
Seifert surfaces of fibered knots
The answer is "yes". See Corollary 2 on page 119 of Thurston's article A norm for the homology of 3-manifolds.
In fact, fibered knots fibre in exactly one way. And all minimal genus Seifert surfaces …
- 28.2k
7
votes
Accepted
The complement of a properly embedded annulus in a handlebody is a handlebody
The answer to the question, as asked, is "no".
For, suppose that $B$ is a three-ball. So, $B$ is a genus zero handlebody. Let $\alpha$ be a knotted arc properly embedded in $B$. So the fundamental g …
- 28.2k
5
votes
Accepted
$\partial$-incompressibility of a surface obtained when attaching a 2-handle to an irreducib...
Suppose that $S$, the relevant boundary component of $M$, is a torus. Suppose that $G$ is the given essential two-sphere in the filled manifold $N$. We isotope $G$ to have minimal intersection with …
- 28.2k
6
votes
Accepted
Is it possible to fill a boundary component of an irreducible 3-manifold using a handlebody ...
EDIT: Here is a substantial rewrite of my previous (incomplete) answer. I think that this proof is a bit "heavy", but I haven't yet thought of a better approach.
The answer is "yes". We split into …
- 28.2k
11
votes
Accepted
What is the minimal genus of a surface acted on by the symmetric group $S_n$?
I don't have a precise answer, but the genus of $S$ has to grow like $n!$.
To see this, note that when $n$ is large enough, $S$ cannot be the sphere or torus. So $S$ admits hyperbolic metrics. By Ni …
- 43.2k
7
votes
Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which ...
Yes. This follows from the characterisation of boundary slopes as a (union of) Lagrangian subspaces of the “symplectic space” of all slopes. See Theorem 1 of Hatcher’s paper.
- 28.2k
2
votes
If the complement of a knot $K$ fibers over the circle is $K$ necessarily fibered?
Suppose that $K$ is a smooth knot in the three-sphere $S^3$. We set $X = S^3 - K$. Suppose that $p \colon X \to S^1$ is a smooth fiber bundle. Let $F_z = p^{-1}(z)$ be the fiber lying over $z$. Ex …
- 28.2k
13
votes
Accepted
Detecting a PL sphere and decompositions
To answer question 1:
There are fast (and easy to implement) algorithms to recognize the zero-, one-, and two-dimensional spheres.
Recognising the three-sphere is fast in practice. An algorithm to do …
- 28.2k
12
votes
Accepted
Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ ?
An incompressible and boundary incompressible connected surface is isotopic to either (a) a vertical annulus or (b) a horizontal surface. A vertical annulus is of the form $\alpha \times I$ where $\a …
- 28.2k
2
votes
On the history of cone-3-manifolds
Here is earlier, but related, work on building manifolds out of geometric polyhedra:
In his thesis [1912] Gieseking builds a manifold $M$ by identifying the faces of a single regular ideal tetrahedron …
- 28.2k
1
vote
Accepted
Figure 8 knot incomplete hyperbolic structure
The fixed points of
$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$, acting on $\mathbb{CP}^1$, are found by solving, for $z$ and $w$, the equation $azw + bw^2 = cz^2 + dzw$. Much of the time you can …
- 28.2k