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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).
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
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
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
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 …
- 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
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 …
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17
votes
Accepted
Homology generated by lifts of simple curves
As far as I know, this is open.
In fact, I think the following weaker question is open.
Let $\Theta$ be the set of loops $\gamma$ in $\widetilde \Sigma$ such that the image of $\gamma$ in $\Sigma$ …
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6
votes
Accepted
Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?
An end of an orientable finite volume hyperbolic $3$--manifold always has a neighborhood homeomorphic to $S^1 \times S^1 \times \mathbb{R}$, so no. Introductory texts on hyperbolic manifolds will con …
- 10.6k
26
votes
Accepted
Diffeomorphisms vs homeomorphisms of 3-manifolds
$\mathsf{Diff}(S^3)$ is homotopy equivalent to $O(4)$, by Hatcher's solution to the Smale Conjecture: Hatcher, Allen E. A proof of the Smale conjecture, Diff(S3)≃O(4). Ann. of Math. (2) 117 (1983), no …
- 10.6k
7
votes
Accepted
Stallings fibration theorem
This should follow from geometrization. The fundamental group of the manifold is $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}$, and geometrization should tell you that the manifold is then $\mathrm{RP}^ …
- 10.6k
8
votes
Cohomology of representation varieties
It think it is difficult to say anything in general, but I would suggest starting by looking at the work of W. Goldman to get a feel for the subject.
For instance, even computing $H^0(\mathrm{Hom}(\p …
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