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Results tagged with gt.geometric-topology
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user 11142
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
0
answers
129
views
Images of the $3$-dimensional solvable geometry
This is more of a mathematical art question. Most people are familiar with the mathematical movie "Not Knot", which explores a hyperbolic $3$-manifold. My question is: are there any images along the s …
- 96.4k
6
votes
2
answers
395
views
pseudo-Anosovs with given action on homology
It is well-known that the Mapping Class Group of a closed surface of genus $g$ surjects onto $Sp(2g, \mathbb{Z})$ (see, for example the Farb-Margalit book). However, I was wondering if there is a simp …
- 96.4k
5
votes
0
answers
244
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Does every null-homologous surface bound, part deux
This is a follow-up to this question, and Mark Grant's excellent answer. The answer shows that the answer is yes, but what if one were under a strange compunction to actually construct the submanifold …
- 96.4k
1
vote
0
answers
237
views
Stupid terminological question on mapping class groups
There is a standard map $T: Mod(S) \rightarrow Sp(2g, \mathbb{Z})$ from the mapping class group of a surface to the symplectic group (where a mapping class goes to its action on the first homology gro …
- 96.4k
8
votes
2
answers
1k
views
Teichmuller volume of moduli space
Someone asked me this question, and I was embarrassed to not know the answer: is the volume of Moduli space with respect to the Teichmuller metric finite? The answer is "yes" when we replace Teichmull …
- 96.4k
6
votes
1
answer
1k
views
Stallings fibration theorem
Stallings' fibration theorem states that if we have a compact irreducible $3$-manifold $M^3,$ with
$G\rightarrow \pi_1(M^3) \rightarrow \mathbb{Z},$ and $G$ is finitely generated and is not of order …
- 96.4k
13
votes
4
answers
930
views
Translation distance in the curve complex
Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the t …
- 96.4k
5
votes
0
answers
142
views
List of cubical spaces
Suppose I have a three-dimensional cube (I tend to think of it as a regular ideal cube in $\mathbb{H}^3,$ but you don't have to). I glue up its sides in some way to obtain topological spaces. The ques …
- 96.4k
8
votes
1
answer
181
views
Hakenness of Heegard splitting
This is somewhat related to this previous quesiton. Suppose I give you a Heegard splitting of $M^3$ of genus $g$ with a gluing map $\phi.$ Is there some condition on $\phi$ which would guarantee that …
- 96.4k
12
votes
1
answer
416
views
Which compact 3-manifolds with boundary embed in $\mathbb{S}^3.$
This is a more sensible (IMHO) restatement of this question:
Which Compact $3$-manifolds with boundaries embed in $\mathbb{S}^3?$ Is there any hope of a characterization?
- 96.4k
7
votes
2
answers
685
views
Infinitely generated Kleinian groups
Is it true that the Hausdorff dimension of the limit set of a Kleinian group is the supremum of Hausdorff dimensions of finitely generated subgroups [perhaps under addtional hypotheses?]I can't seem t …
- 96.4k
9
votes
1
answer
509
views
Hyperbolic 3-manifolds fibering over the circle
Suppose you have the mapping torus $M_\phi$ of some pseudo-Anosov map $\phi.$ Is there some sufficient or necessary condition on $\phi$ to assure that $M_\phi$ has large injectivity radius? I am aware …
- 96.4k
1
vote
0
answers
86
views
Dirichlet domain complexity
Suppose I have a subgroup of $SL(2, \mathbb{Z})$ given by explicit matrix generators, and my goal in life is to construct a Dirichlet domain (for, say, everyone's favorite basepoint $\sqrt{-1}$). Is t …
- 96.4k
2
votes
1
answer
280
views
Unit tangent bundles of principal congruence orbifolds
In this question it is noted that $SL(2, \mathbb{R})/SL(2, \mathbb{Z})$ is homeomorphic to the trefoil complement in $S^3.$ Is there a similarly nice interpretation of $SL(2, \mathbb{R})/\Gamma(N)$ fo …
- 96.4k
6
votes
1
answer
300
views
Generating prime knots (in order)
In this really cool paper https://arxiv.org/abs/1612.03368, A. Malyutin shows that the probability that a random prime knot of up to $N$ crossings (as $N$ goes to infinity) is not generically hyperbol …
- 96.4k