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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).
5
votes
A strong annulus theorem for 3-manifolds
I think that Theorem 15.1.5 in Scott's unpublished book on three-manifolds (email him for a copy) is very close to what you want.
- 28.2k
3
votes
pseudo-Anosovs with given action on homology
Suppose that $f$ is a pseudo-Anosov and let $\lambda^\pm$ be its stable and unstable laminations. Suppose that $g$ is any mapping class with the following property: $g(\lambda^+) \neq \lambda^-$. Th …
- 28.2k
4
votes
Accepted
Does a regular neighborhood always exist for a properly embedded surface in a 3-manifold?
The answer is yes, with the correct technical hypothesis of "local flatness". (Local flatness rules out, for example, the sort of behavior shown by the Alexander horned sphere.) You are correct to th …
- 28.2k
0
votes
What kind of 3-manifolds arise has hypersurfaces in R^4?
You can cut a two-torus out of $\mathbb{R}^3$ and I think that a simple modification will cut a three-torus out of $\mathbb{R}^4$. So the answer to your second question is "yes".
- 28.2k
2
votes
Wide cylinders on half-translation surfaces
By passing to the branched double cover, branched over all singular points, it suffices to address your question for translation surfaces. (This gives up at worst a factor of two from the width.)
Som …
- 28.2k
4
votes
Accepted
Factoring maps of handlebodies
Suppose that we are given a PL map from a handlebody $W$ to a handlebody
$V$. Choose a spine for $W$. Homotope the map until the image is a
regular neighborhood of the image of the spine. By genera …
- 28.2k
12
votes
Accepted
Finding hyperbolic metrics by approximation
To my knowledge this hasn't been done in theory (although see Harriet Moser's thesis http://www.math.columbia.edu/~moser/). But it certainly has been done in practice by Jeff Weeks' program SnapPea. …
- 28.2k
6
votes
Accepted
Sufficient conditions for a 3D tetrahedral complex to be homeomorphic to a 3D ball
$\newcommand{\RR}{\mathbb{R}}$The other answers are completely general, but there is simpler way if we use the (given) hypothesis that all the action is taking place in $\RR^3$. So, suppose that $T$ …
- 28.2k
3
votes
Accepted
Hyperbolic links
There is not enough information in your picture to give a definitive answer. If you have more details about the link $L$ then an answer will probably factor through Thurston's characterisation of hyp …
- 28.2k
5
votes
Accepted
Extending a 2-complex embedded in $\mathbb{S}^3$ into a simply connected one
Yes. [The tameness of $f$ makes our life much easier.]
It will be convenient for the proof to assume that $C$ is pure: that is, every vertex and edge of $C$ lies in some triangle of $C$. If this is n …
- 28.2k
2
votes
Properly embedded annuli in genus two handlebody?
Here is another construction: Let $S$ be the once-holed torus: that is, a two-torus minus a small open disk. Then $V = S \times [0, 1]$ is homeomorphic to a genus two handlebody. [Exercise.]
Let $\al …
- 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
3
votes
Accepted
Immersed quasi-Fuchsian surfaces surviving Dehn fillings
Notice that Hatcher proves that three-manifolds with a single torus boundary have only finitely many embedded boundary slopes. So I assume that your question one is asking about "immersed boundary sl …
- 28.2k
2
votes
Accepted
Is the following 3-manifold always a trivial I-bundle over a surface?
This follows, fairly easily, from the hypothesis of irreducibility and from the "annulus theorem" (see page 130 of Jaco-Shalen's book "Seifert Fibered Spaces in 3-Manifolds"). You can remove the hypo …
- 28.2k
7
votes
Accepted
Refining a triangulation
Suppose that $S$ is a closed, connected surface with negative Euler characteristic. Suppose that $T$ is a triangulation of $S$.
Define "refine" to mean "replace each triangle by four triangles" (so t …
- 28.2k