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Results tagged with gt.geometric-topology
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user 1335
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).
29
votes
Fundamental group of 3-manifold with boundary
No. The Baumslag--Solitar groups $\langle a, b | ab^m a^{-1} = b^n \rangle$ are not three-manifold groups when $m \neq n$.
See:
Heil, Wolfgang H. Some finitely presented non-$3$-manifold groups. Proc …
- 28.2k
8
votes
Uniqueness of a polygon
The moduli space of $n$-gons up to orientation preserving similarity can be identified with $\mathbb{C}P^{n-2}$, so I would say $2n-3$.
See On the moduli space of polygons in the Euclidean plane by Ka …
- 2,821
9
votes
Accepted
Braid groups acting on CAT(0)-complexes
As Sam Nead says, $B_n$ contains $\mathbb{Z}^2 * \mathbb{Z}$, and you can find an example the way he suggests.
If you'd like something much more explicit, you can simply take the first three standard …
- 35.5k
14
votes
Torsion in homology or fundamental group of subsets of Euclidean 3-space
I'll assume that the subset is compact.
Then, if you use Cech cohomology, Alexander duality turns this into a question about the complement, which is a 3-manifold.
So, I answer with another question: …
- 4,713
4
votes
SL(2, C)-representation of a knot
This is a good introduction:
Shalen, Representations of 3-manifold groups. Handbook of geometric topology, 955–1044, North-Holland, Amsterdam, 2002.
- 10.6k
12
votes
Accepted
Examples of acylindrical 3-manifolds
(source)
The exterior of Suzuki's Brunnian graph on $n$-edges, here pictured with $n=7$, is irreducible, atoroidal, boundary incompressible, and acylindrical. See
Luisa Paoluzzi and Bruno Zi …
- 1,228
15
votes
Accepted
Word problem for fundamental group of submanifolds of the 4-sphere
Update:
My memory was quite blurry about this when I originally answered.
See Gonzáles-Acuña, Gordon, Simon, ``Unsolvable problems about higher-dimensional knots and related groups,'' L’Enseigneme …
- 10.6k
21
votes
Accepted
For which surfaces is Penner's conjecture known to be true?
Shin and Strenner have shown that the conjecture is false when 3g + n > 4.
See http://arxiv.org/abs/1410.6974
- 10.6k
14
votes
Accepted
Do different Dehn fillings produce homeomorphic 3-manifolds ?
This phenomenon is called "cosmetic surgery."
If $K$ is an amphichiral knot in the $3$--sphere with exterior $M_K$, then $M_K(p/q) \cong - M_K(-p/q)$. So if $p/q$ is a hyperbolic filling slope, the …
- 10.6k
4
votes
Translation distance in the curve complex
In the braid group, Ko and Lee have given a polynomial time test of reducibility using the Garside structure. (See http://arxiv.org/abs/math/0610746)
- 10.6k
11
votes
Accepted
Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?
Yes. Every (compact) flat $n$-manifold is diffeomorphic to a cusp cross section of a hyperbolic $(n+1)$-manifold. This is a theorem of McReynolds, Controlling manifold covers of orbifolds, Math. Res. …
- 10.6k
13
votes
Accepted
Hyperbolic structures on $S\times\mathbb{R}$
It follows from Thurston's Covering Theorem that there are no such examples.
The covering theorem says that if a degenerate end is infinite-to-one under a covering map, then you are (virtually) in th …
- 10.6k
2
votes
Accepted
Does the fundamental group of a surface have rigid subgroups?
Regarding Question 2, you get lots of examples that are rigid for the lifting map $M(\Gamma) \to M(\Gamma_B)$.
Let $B$ be finitely generated subgroup of $\Gamma$ (considered a fuchsian group) such t …
- 10.6k
6
votes
Why should I care about Heegaard-Floer theory?
Kronheimer, Mrowka, Ozsváth, and Szabó obtained a new proof of Gordon and Luecke's Knot Complement Theorem using monopole Floer homology. That's pretty good, I think. They also proved that $\mathbb{ …
- 10.6k
10
votes
Higher homotopy groups of slice disk complement
To add to Ryan's answer,
$2$--knots usually don't have aspherical complements, see Dyer & Vasquez, The sphericity of higher dimensional knots, Canad. J. Math. 25(1973), 1132-1136. This suggests a com …
- 10.6k