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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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2
answers
2k
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Conformal embedding of Riemann surfaces into 3-space
I can't seem to find any work on the following question: Can every (closed, of finite type) Riemann surface $S$ be realized as an embedded (or even immersed) smooth surface in Euclidean $3$-space, whe …
- 96.4k
23
votes
7
answers
9k
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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 …
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 …
- 96.4k
18
votes
6
answers
6k
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Computing signature
I have a feeling that this might have already been asked, but can't find the question. Anyway, the question is: given a symmetric $n\times n$ matrix, is there a faster way to compute its signature tha …
- 96.4k
13
votes
3
answers
1k
views
A four-dimensional counterexample?
Does anyone know an example of a smooth hyperbolic surface bundle over a hyperbolic surface (surface = compact two-manifold) which does not have a complex structure? Is there any decision procedure to …
- 96.4k
13
votes
2
answers
2k
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Altitudes of a triangle
The three altitudes of a triangle are concurrent -- this is true in all three constant curvature geometries (Euclidean, hyperbolic, spherical), but, as far as I know, the proofs are different in the t …
- 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
13
votes
0
answers
370
views
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
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?
- 96.4k
11
votes
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 …
- 96.4k
11
votes
1
answer
327
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Rank and hyperbolic volume
Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of Cu …
- 96.4k
11
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3
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1k
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orbifold covering
Given two compact surfaces $S_1$ and $S_2$ of genus at least $2,$ it is easy to tell when $S_1$ covers $S_2:$ whenever $\chi(S_2)$ divides $\chi(S_1).$ Now, suppose I have two orbifolds of negative Eu …
- 96.4k
9
votes
1
answer
509
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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
8
votes
2
answers
1k
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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
8
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2
answers
1k
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Homological computations
Suppose I have a group acting on some Hadamard manifold, and I want to understand as much as possible about the (co)homology of the quotient. In my case I can find a fundamental domain for the action …
- 96.4k