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Results tagged with gt.geometric-topology
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user 20787
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
Jordan curve theorem for cylinders
If you are willing to quote the Schoenflies theorem and the classification of surfaces, a quick proof of this result is a standard exercise. First, using algebraic topology (as in the proof of the Jor …
- 2,858
9
votes
Accepted
Nielsen-Thurston classification via the curve complex?
Masur and Minsky, in their paper "Quasiconvexity in the curve complex", give a purely combinatorial proof that certain "nested" train track sequences project to quasigeodesics in the curve complex. It …
- 28.2k
3
votes
Accepted
A closed curve can be homotopic to remove all intersections with a filling $\Gamma$ if it ha...
This is true, here's a proof, by a kind of "Whitney trick".
Perturb the set of curves $\{\gamma_i\} \cup \{c\}$ to put it into general position, so they are pairwise transverse and there is no triple …
- 15.4k
3
votes
Accepted
About isotopy and homotopy
Once you have found an annulus $R \subset S$ whose two boundary components are $\alpha$ and $\beta$, by definition of "annulus" there exists a homeomorphism $H : S^1 \times [0,1] \to R$. The compositi …
- 15.4k
11
votes
Can we determine which monodromy of surface gives a fibered knot?
Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=\mathbb{Z}$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce …
- 1,430
4
votes
Accepted
Cusps of hyperbolic surfaces under finite covers
Assume just that $\Gamma$ has index $k$ in $\Gamma'$. Let $C \subset \mathbb R \cup \{\infty\}$ be the set of parabolic points for the action of $\Gamma$. Then $C$ is also the set of parabolic points …
- 15.4k
3
votes
Accepted
Is transverse measure on a foliation without closed leaves unique?
Such foliations were studied rather intensely in the early works on measured foliations that introduced them to the mathematical world. See for example "Thurston's work on surfaces" aka "Travaux de Th …
- 15.4k
8
votes
What is a geodesic in Outer space?
Besides the geodesic paths of the asymmetric metric $d(\cdot,\cdot)$ that are mentioned in other answers (namely paths such that $d(\gamma(s),\gamma(t)) = t-s$ if $s \le t$), there is another class of …
- 15.4k
2
votes
Accepted
Classifying transverse curves to a surface foliation carried by a train track
The train track $\tau$ has a dual bigon track $\tau^\perp$, described in (for example) Penner's book. The bigon track $\tau^\perp$ might not be maximal, but it can be enlarged in various ways by trian …
- 15.4k
10
votes
Accepted
Do regular points of an orbifold form a connected set?
Part of the reason that people consider orbifolds whose singular points have codimension $\ge 2$ is because they are restricting their attention to oriented orbifolds. For example, if you are working …
- 15.4k
6
votes
Accepted
Can every curve be made transversal to a foliation by applying a pseudo-Anosov?
Given a full support measured foliation $\mathcal F$ and given a pseudo-Anosov $\phi$, what you want will work as long as $\mathcal F^u_\phi$ can be isotoped to be transverse to $\mathcal F$.
The tro …
- 15.4k
3
votes
Accepted
Are isotopic transversal curves on a foliated surface transversally isotopic?
"Reebless" is not strong enough for this to be true. One should also assume that there is no "half-Reeb" annulus, i.e. for every circle leaf $C$ and for each side of $C$ the nearby leaves on that side …
- 15.4k
5
votes
Accepted
Length of a simple closed curve under Pseudo-Anosov maps
To answer the main question as well as the question in the comments, for every simply closed curve $a$ we have $I(\mu^s(a)) \in (0,\infty)$, and every $n$ we have $I(\mu^s(f^n(a))) = \lambda^{-n} I(\m …
- 15.4k
6
votes
Accepted
Are pseudo-Anosov foliations dense?
The pseudo-Anosov foliations form a subset of $\mathcal{PMF}(F)$ which is invariant under the action of the mapping class group $MCG(F)$, because if $\Lambda_+(\phi) \in \mathcal{PMF}(F)$ is the stabl …
- 15.4k
5
votes
Accepted
Does there exist a closed geodesic go through a $\epsilon$-net of a hyperbolic surface?
Yes, such a closed geodesic always exists. See Theorem 1.1 of this paper by Basmajian, Parlier, and Souto (which I found by searching under the term "density of closed geodesics on a hyperbolic surfac …
- 15.4k