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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).
23
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7
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Introduction to Floer Theory?
Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive t …
- 4,713
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 …
- 68.9k
5
votes
1
answer
374
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Extension of covering map
The question is the following: suppose I have manifolds $N$ and $M$ both with boundary, and I have a covering map $\phi$ from $\partial N$ to $\partial M.$ The question is: when is there a covering ma …
- 68.9k
7
votes
1
answer
807
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More four-dimensional counterexamples
To follow up on A four-dimensional counterexample?, I am probably being dense, but are there examples of spaces which are homotopy equivalent to bundles of surfaces over surfaces (or three-manifolds o …
- 12.8k
11
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1
answer
253
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Algorithmic Borel finiteness for hyperbolic manifolds
It is a theorem of Borel that there is a finite number of arithmetic hyperbolic manifolds of volume bounded above by $V.$ Is there any algorithm (or hope of an algorithm) to actually construct all of …
- 63.9k
5
votes
3
answers
522
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Triangulations of 3-manifolds in Regina and SnapPy
I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a …
- 656
20
votes
4
answers
3k
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Simply-connected rational homology spheres
Every simply-connected rational homology sphere is, in fact, the usual sphere in dimensions $2, 3.$ Is this true in dimension 4? Where are the first counterexamples? (I know there are some in dimensio …
- 28k
7
votes
1
answer
355
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Number of (distinct) knots with a bounded number of crossings
The title pretty much covers it: are there good (asymptotic) estimates on the number of knot types whose projection has at most $N$ crossings? Similar question with "projection" replaced by "alternati …
- 68.9k
6
votes
1
answer
300
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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
12
votes
1
answer
416
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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?
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13
votes
0
answers
370
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Euler characteristic of *hyperbolic* orbifolds
This is a follow-up to this question. Which rational numbers arise as Euler characteristics of orbifold quotients of $\mathbb{H}^n?$ The answer is not even clear for $n=2.$ It is clear that the Euler …
- 96.4k
1
vote
0
answers
86
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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
5
votes
0
answers
129
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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 …
- 63.9k
6
votes
2
answers
654
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fundamental groups of curves
I saw the following statement made without proof in a paper of Bogomolov and Tschinkel:
If $X$ is an algebraic surface, and $C$ is an ample smooth curve in $X,$ then the fundamental group of $C$ surj …
- 11.2k
4
votes
1
answer
504
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Quotient of principal congruence subgroups
This is a direct follow-up to this question. What is the quotient $\Gamma(2)/\Gamma(2^n)?$ (the principal congruence subgroups are in $SL(2, \mathbb{Z}).$ It is a 2-group, but what else?
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