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Results tagged with gt.geometric-topology
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user 1463
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).
10
votes
Finding hyperbolic metrics by approximation
You might be interested in this paper of Jason Manning. He proves (assuming the solution to the word problem in $G$---basic undecidability results tell you that it's necessary to assume something lik …
- 25.2k
8
votes
Accepted
Attaching a thickened annulus between two 3-manifold
The answer is 'yes'. Furthermore, this is more or less equivalent to a group-theoretic fact, which applies in much greater geneality, called Shenitzer's Lemma.
First, note that we may assume that $X …
- 25.2k
8
votes
The homeomorphism problem for hyperbolic 3-manifolds and the virtual Haken theorem
The trouble is that computing the outer automorphism group is a very similar to the isomorphism problem, but actually a little harder. The only algorithm that I know of to compute the outer automorph …
- 25.2k
5
votes
Accepted
Covering spaces of surfaces
Given any covering map $\Sigma_h\to\Sigma_g$ between two surfaces, is there some kind of a ``standard" covering $M\to \Sigma_g$, which factors through $\Sigma_h$?
In brief, the answer to this pa …
- 25.2k
6
votes
Accepted
Link of a vertex of a 3-orbifold (link orbifold)
Let $O$ be the 3-orbifold, $\widetilde{O}$ its universal cover (it sounds like we're assuming that $O$ is 'good', ie has a manifold universal cover) and $\pi_1O$ its fundamental group.
Let $v$ be a v …
- 25.2k
2
votes
Example of Noetherian group (every subgroup is finitely generated) that is not finitely pres...
It's unknown whether every slender group is virtually polycyclic. See page 87 of Matt Clay's thesis.
EDIT: Primoz rightly points out that a Tarski monster is slender (and not finitely presentable!). …
- 25.2k
9
votes
Why should I care about Heegaard-Floer theory?
Yi Ni used Heegaard Floer Homology to prove, among many other things, that a knot admitting a lens-space surgery is fibred. I believe that no 'conventional' proof of this is known.
Ni, Yi, Knot Floe …
3
votes
Accepted
On the realization of a quotient group
For a finite polyhedron $P$ and finite-index normal subgroup $N$ of $G=\pi_1P$, there is a canonical finite polyhedron $Q$ with $\pi_1Q\cong G/N$ constructed as follows. Let $\tilde{P}\stackrel{p}{\t …
- 25.2k
9
votes
Accepted
Asphericity of 2-complexes
The answer is no.
This follows from a theorem of Collins and Miller, who constructed a recursive sequence of presentations $P_n$ such that the set of $n$ for which $P_n$ presents the trivial group is …
- 25.2k
1
vote
A simple closed curve on a surface
This paper of Boggi gives an attractive algebraic characterisation of the conjugacy classes of the fundamental group $\pi_1\Sigma$ that are represented by simple closed curves. The idea is nice and ea …
- 25.2k
5
votes
Accepted
Angles and Busemann function in CAT(0)
The answer is "no" even in the hyperbolic plane $\mathbb{H}^2$.
Consider the horocycle about $\xi$ through $x$: this is the set of points such that $b_{\xi}(z)=0$. Let $\gamma$ be the geodesic throug …
- 25.2k
8
votes
Accepted
Difficulty with "On fibering certain 3-manifolds" by Stallings
I think about it like this. For convenience, I'll assume $M$ is closed.
Given a homomorphism $\phi:\pi_1M\to\mathbb{Z}$, Stallings explains how to find an essential surface $S\subset M$ with $\pi_1S\ …
- 25.2k
39
votes
finite generated group realized as fundamental group of manifolds
Theorem. Every finitely presentable group is the fundamental group of a closed 4-manifold.
Sketch proof. Let $\langle a_1,\ldots,a_m\mid r_1,\ldots, r_n\rangle$ be a presentation. By van Kampen, th …
- 25.2k
13
votes
Accepted
Does every retraction of free groups arise from projection to a subset of a freely generatin...
No. This is explicitly stated in the paragraph above Theorem 1 of:
Turner, Edward C, Test words for automorphisms of free groups.
Bull. London Math. Soc. 28 (1996), no. 3, 255--263.
The author refer …
- 25.2k
17
votes
When is a finitely generated group finitely presented?
An often-used method is to compute $H_2$. If the group is finitely presentable then $H_2$ is of finite rank with any coefficients.
For instance, you can use this technique to show that if $q:F\to\ma …